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RD Sharma solutions for Class 11 Mathematics chapter 9 - Values of Trigonometric function at multiples and submultiples of an angle

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RD Sharma Mathematics Class 11

Mathematics Class 11

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Ex. 9.1Ex. 9.2Ex. 9.3Others

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle Exercise 9.1 solutions [Pages 28 - 30]

Ex. 9.1 | Q 1 | Page 28

Prove that:  \[\sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}} = \tan x\]

Ex. 9.1 | Q 2 | Page 28

Prove that:  \[\frac{\sin 2x}{1 - \cos 2x} = cot x\]

Ex. 9.1 | Q 3 | Page 28

Prove that:  \[\frac{\sin 2x}{1 + \cos 2x} = \tan x\]

 
Ex. 9.1 | Q 4 | Page 28

Prove that: \[\sqrt{2 + \sqrt{2 + 2 \cos 4x}} = 2 \text{ cos } x\]

 
Ex. 9.1 | Q 5 | Page 28

Prove that:  \[\frac{1 - \cos 2x + \sin 2x}{1 + \cos 2x + \sin 2x} = \tan x\]

 
Ex. 9.1 | Q 6 | Page 28

Prove that:  \[\frac{\sin x + \sin 2x}{1 + \cos x + \cos 2x} = \tan x\]

 
Ex. 9.1 | Q 7 | Page 28

Prove that:  \[\frac{\cos 2 x}{1 + \sin 2 x} = \tan \left( \frac{\pi}{4} - x \right)\]

 
Ex. 9.1 | Q 8 | Page 28

Prove that:  \[\frac{\cos x}{1 - \sin x} = \tan \left( \frac{\pi}{4} + \frac{x}{2} \right)\]

Ex. 9.1 | Q 9 | Page 28

Prove that: \[\cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} + \cos^2 \frac{5\pi}{8} + \cos^2 \frac{7\pi}{8} = 2\]

Ex. 9.1 | Q 10 | Page 28

Prove that: \[\sin^2 \frac{\pi}{8} + \sin^2 \frac{3\pi}{8} + \sin^2 \frac{5\pi}{8} + \sin^2 \frac{7\pi}{8} = 2\]

Ex. 9.1 | Q 11 | Page 28

Prove that: \[\left( \cos \alpha + \cos \beta^2 \right) + \left( \sin \alpha + \sin \beta \right)^2 = 4 \cos^2 \left( \frac{\alpha - \beta}{2} \right)\]

 
Ex. 9.1 | Q 12 | Page 28

Prove that:  \[\sin^2 \left( \frac{\pi}{8} + \frac{x}{2} \right) - \sin^2 \left( \frac{\pi}{8} - \frac{x}{2} \right) = \frac{1}{\sqrt{2}} \sin x\]

 
Ex. 9.1 | Q 13 | Page 28

Prove that: \[1 + \cos^2 2x = 2 \left( \cos^4 x + \sin^4 x \right)\]

 
Ex. 9.1 | Q 14 | Page 28

Prove that: \[\cos^3 2x + 3 \cos 2x = 4\left( \cos^6 x - \sin^6 x \right)\]

 
Ex. 9.1 | Q 15 | Page 28

Prove that: \[\left( \sin 3x + \sin x \right) \sin x + \left( \cos 3x - \cos x \right) \cos x = 0\]

Ex. 9.1 | Q 16 | Page 28

Prove that: \[\cos^2 \left( \frac{\pi}{4} - x \right) - \sin^2 \left( \frac{\pi}{4} - x \right) = \sin 2x\]

Ex. 9.1 | Q 17 | Page 28

Prove that:  \[\cos 4x = 1 - 8 \cos^2 x + 8 \cos^4 x\]

 

Ex. 9.1 | Q 18 | Page 28

Prove that: \[\sin 4x = 4 \sin x \cos^3 x - 4 \cos x \sin^3 x\]

 
Ex. 9.1 | Q 19 | Page 28

Show that: \[3 \left( \sin x - \cos x \right)^4 + 6 \left( \sin x + \cos \right)^2 + 4 \left( \sin^6 x + \cos^6 x \right) = 13\]

 
Ex. 9.1 | Q 20 | Page 28

Show that: \[2 \left( \sin^6 x + \cos^6 x \right) - 3 \left( \sin^4 x + \cos^4 x \right) + 1 = 0\]

 
Ex. 9.1 | Q 21 | Page 28

Prove that: \[\cos^6 A - \sin^6 A = \cos 2A\left( 1 - \frac{1}{4} \sin^2 2A \right)\]

 
Ex. 9.1 | Q 22 | Page 28

Prove that:\[\tan\left( \frac{\pi}{4} + x \right) + \tan\left( \frac{\pi}{4} - x \right) = 2 \sec 2x\]

 
Ex. 9.1 | Q 23 | Page 28

Prove that: \[\cot^2 x - \tan^2 x = 4 \cot 2 x  \text{ cosec }  2 x\]

 
Ex. 9.1 | Q 24 | Page 28

Prove that: \[\cos 4x - \cos 4\alpha = 8 \left( \cos x - \cos \alpha \right) \left( \cos x + \cos \alpha \right) \left( \cos x - \sin \alpha \right) \left( \cos x + \sin \alpha \right)\]

Ex. 9.1 | Q 25 | Page 28

Prove that \[\sin 3x + \sin 2x - \sin x = 4 \sin x \cos\frac{x}{2} \cos\frac{3x}{2}\]

Ex. 9.1 | Q 26 | Page 29
\[\tan 82\frac{1° }{2} = \left( \sqrt{3} + \sqrt{2} \right) \left( \sqrt{2} + 1 \right) = \sqrt{2} + \sqrt{3} + \sqrt{4} + \sqrt{6}\]

 

Ex. 9.1 | Q 27 | Page 29

Prove that: \[\cot \frac{\pi}{8} = \sqrt{2} + 1\]

 
Ex. 9.1 | Q 28.1 | Page 29

 If \[\cos x = - \frac{3}{5}\]  and x lies in the IIIrd quadrant, find the values of \[\cos\frac{x}{2}, \sin\frac{x}{2}, \sin 2x\] .

 

 

Ex. 9.1 | Q 28.2 | Page 29

 If  \[\cos x = - \frac{3}{5}\]  and x lies in IInd quadrant, find the values of sin 2x and \[\sin\frac{x}{2}\] .

 

 

Ex. 9.1 | Q 29 | Page 29

If  \[\sin x = \frac{\sqrt{5}}{3}\] and x lies in IInd quadrant, find the values of \[\cos\frac{x}{2}, \sin\frac{x}{2} \text{ and }  \tan \frac{x}{2}\] . 

 

 

Ex. 9.1 | Q 30.1 | Page 29

 If 0 ≤ x ≤ π and x lies in the IInd quadrant such that  \[\sin x = \frac{1}{4}\]. Find the values of \[\cos\frac{x}{2}, \sin\frac{x}{2} \text{ and }  \tan\frac{x}{2}\]

 

 

Ex. 9.1 | Q 30.2 | Page 29

 If \[\cos x = \frac{4}{5}\]  and x is acute, find tan 2

 

Ex. 9.1 | Q 30.3 | Page 29

 If \[\sin x = \frac{4}{5}\] and \[0 < x < \frac{\pi}{2}\]

, find the value of sin 4x.

 

 

Ex. 9.1 | Q 31 | Page 29

If \[\text{ tan } x = \frac{b}{a}\] , then find the value of \[\sqrt{\frac{a + b}{a - b}} + \sqrt{\frac{a - b}{a + b}}\] . 

 

 

Ex. 9.1 | Q 32 | Page 29

If \[\tan A = \frac{1}{7}\]  and \[\tan B = \frac{1}{3}\] , show that cos 2A = sin 4

 

 

Ex. 9.1 | Q 33 | Page 29

Prove that:  \[\cos 7°  \cos 14° \cos 28° \cos 56°= \frac{\sin 68°}{16 \cos 83°}\]

 
Ex. 9.1 | Q 34 | Page 29

Prove that: \[\cos\frac{2\pi}{15} \cos\frac{4\pi}{15} \cos \frac{8\pi}{15} \cos \frac{16\pi}{15} = \frac{1}{16}\]

Ex. 9.1 | Q 35 | Page 29

Prove that: \[\cos\frac{\pi}{5}\cos\frac{2\pi}{5}\cos\frac{4\pi}{5}\cos\frac{8\pi}{5} = \frac{- 1}{16}\]

 
Ex. 9.1 | Q 36 | Page 29

Prove that: \[\cos \frac{\pi}{65} \cos \frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65} = \frac{1}{64}\]

 
Ex. 9.1 | Q 37 | Page 29

If \[2 \tan \alpha = 3 \tan \beta,\]  prove that \[\tan \left( \alpha - \beta \right) = \frac{\sin 2\beta}{5 - \cos 2\beta}\] .

 
Ex. 9.1 | Q 38.1 | Page 29

If \[\sin \alpha + \sin \beta = a \text{ and }  \cos \alpha + \cos \beta = b\] , prove that 
(i)\[\sin \left( \alpha + \beta \right) = \frac{2ab}{a^2 + b^2}\]

Ex. 9.1 | Q 38.2 | Page 29

If \[\sin \alpha + \sin \beta = a \text{ and }  \cos \alpha + \cos \beta = b\] , prove that

(ii) \[\cos \left( \alpha - \beta \right) = \frac{a^2 + b^2 - 2}{2}\]

 

Ex. 9.1 | Q 39 | Page 29

If \[2 \tan\frac{\alpha}{2} = \tan\frac{\beta}{2}\] , prove that \[\cos \alpha = \frac{3 + 5 \cos \beta}{5 + 3 \cos \beta}\]

 

 

Ex. 9.1 | Q 40 | Page 29

If \[\cos x = \frac{\cos \alpha + \cos \beta}{1 + \cos \alpha \cos \beta}\] , prove that \[\tan\frac{x}{2} = \pm \tan\frac{\alpha}{2}\tan\frac{\beta}{2}\]

 
Ex. 9.1 | Q 41 | Page 29

If  \[\sec \left( x + \alpha \right) + \sec \left( x - \alpha \right) = 2 \sec x\] , prove that \[\cos x = \pm \sqrt{2} \cos\frac{\alpha}{2}\]

 
Ex. 9.1 | Q 42 | Page 30

If \[\cos \alpha + \cos \beta = \frac{1}{3}\]  and sin \[\sin\alpha + \sin \beta = \frac{1}{4}\] , prove that \[\cos\frac{\alpha - \beta}{2} = \pm \frac{5}{24}\]

 
 

 

Ex. 9.1 | Q 43 | Page 30

If  \[\sin \alpha = \frac{4}{5} \text{ and }  \cos \beta = \frac{5}{13}\] , prove that \[\cos\frac{\alpha - \beta}{2} = \frac{8}{\sqrt{65}}\]

 
Ex. 9.1 | Q 44.1 | Page 30

If \[a \cos2x + b \sin2x = c\]  has α and β as its roots, then prove that 

(i) \[\tan\alpha + \tan\beta = \frac{2b}{a + c}\]

 

Ex. 9.1 | Q 44.2 | Page 30

If \[a \cos2x + b \sin2x = c\]  has α and β as its roots, then prove that

(ii)  \[\tan\alpha \tan\beta = \frac{c - a}{c + a}\]

 

Ex. 9.1 | Q 44.3 | Page 30

If \[a \cos2x + b \sin2x = c\]  has α and β as its roots, then prove that

(iii)\[\tan\left( \alpha + \beta \right) = \frac{b}{a}\] 

 

Ex. 9.1 | Q 45 | Page 30

If  \[\cos\alpha + \cos\beta = 0 = \sin\alpha + \sin\beta\] , then prove that \[\cos2\alpha + \cos2\beta = - 2\cos\left( \alpha + \beta \right)\] .

 

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle Exercise 9.2 solutions [Pages 36 - 37]

Ex. 9.2 | Q 1 | Page 36

Prove that:  \[\sin 5x = 5 \sin x - 20 \sin^3 x + 16 \sin^5 x\]

 
Ex. 9.2 | Q 2 | Page 36

Prove that: \[4 \left( \cos^3 10 °+ \sin^3 20° \right) = 3 \left( \cos 10°+ \sin 2° \right)\]

 
Ex. 9.2 | Q 3 | Page 36

Prove that:  \[\cos^3 x \sin 3x + \sin^3 x \cos 3x = \frac{3}{4} \sin 4x\]

 
Ex. 9.2 | Q 4 | Page 36

\[\tan x \tan\left( x + \frac{\pi}{3} \right) + \tan x \tan\left( \frac{\pi}{3} - x \right) + \tan\left( x + \frac{\pi}{3} \right)\tan\left( x - \frac{\pi}{3} \right) = - 3\]

Ex. 9.2 | Q 5 | Page 36

\[\tan x + \tan\left( \frac{\pi}{3} + x \right) - \tan\left( \frac{\pi}{3} - x \right) = 3 \tan 3x\] 

Ex. 9.2 | Q 6 | Page 36
\[\cot x + \cot\left( \frac{\pi}{3} + x \right) + \cot\left( \frac{\pi}{3} - x \right) = 3 \cot 3x\]

 

Ex. 9.2 | Q 7 | Page 36

\[\cot x + \cot\left( \frac{\pi}{3} + x \right) + \cot\left( \frac{2\pi}{3} + x \right) = 3 \cot 3x\] 

Ex. 9.2 | Q 8 | Page 36
\[\sin 5x = 5 \cos^4 x \sin x - 10 \cos^2 x \sin^3 x + \sin^5 x\]

 

Ex. 9.2 | Q 9 | Page 37
\[\sin^3 x + \sin^3 \left( \frac{2\pi}{3} + x \right) + \sin^3 \left( \frac{4\pi}{3} + x \right) = - \frac{3}{4} \sin 3x\]

 

Ex. 9.2 | Q 10 | Page 37

Prove that \[\left| \sin x \sin \left( \frac{\pi}{3} - x \right) \sin \left( \frac{\pi}{3} + x \right) \right| \leq \frac{1}{4}\]  for all values of x

 
 
Ex. 9.2 | Q 11 | Page 37

Prove that \[\left| \cos x \cos \left( \frac{\pi}{3} - x \right) \cos \left( \frac{\pi}{3} + x \right) \right| \leq \frac{1}{4}\]  for all values of x

 

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle Exercise 9.3 solutions [Page 42]

Ex. 9.3 | Q 1 | Page 42

Prove that: \[\sin^2 \frac{2\pi}{5} - \sin^{2 -} \frac{\pi}{3} = \frac{\sqrt{5} - 1}{8}\]

  
Ex. 9.3 | Q 2 | Page 42

Prove that: \[\sin^2 24°- \sin^2 6° = \frac{\sqrt{5} - 1}{8}\]

  
Ex. 9.3 | Q 3 | Page 42

Prove that:  \[\sin^2 42° - \cos^2 78 = \frac{\sqrt{5} + 1}{8}\] 

 
Ex. 9.3 | Q 4 | Page 42

Prove that:  \[\cos 78°  \cos 42°  \cos 36° = \frac{1}{8}\]

Ex. 9.3 | Q 5 | Page 42

Prove that: \[\cos\frac{\pi}{15}\cos\frac{2\pi}{15}\cos\frac{4\pi}{15}\cos\frac{7\pi}{15} = \frac{1}{16}\]

 
Ex. 9.3 | Q 7 | Page 42

Prove that: \[\cos 6° \cos 42°   \cos 66°    \cos 78° = \frac{1}{16}\]

 
Ex. 9.3 | Q 8 | Page 42

Prove that: \[\cos 36° \cos 42° \cos 60° \cos 78°  = \frac{1}{16}\]

 
Ex. 9.3 | Q 9 | Page 42

Prove that : \[\sin\frac{\pi}{5}\sin\frac{2\pi}{5}\sin\frac{3\pi}{5}\sin\frac{4\pi}{5} = \frac{5}{16}\]

 
Ex. 9.3 | Q 10 | Page 42

Prove that: \[\cos\frac{\pi}{15} \cos \frac{2\pi}{15} \cos \frac{3\pi}{15} \cos \frac{4\pi}{15} \cos \frac{5\pi}{15} \cos\frac{6\pi}{15} \cos \frac{7\pi}{15} = \frac{1}{128}\]

 

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle solutions [Page 42]

Q 1 | Page 42

If \[\cos 4x = 1 + k \sin^2 x \cos^2 x\] , then write the value of k.

 
Q 2 | Page 42

If \[\tan\frac{x}{2} = \frac{m}{n}\] , then write the value of m sin x + n cos x.

 

 

Q 3 | Page 42

If  \[\frac{\pi}{2} < x < \frac{3\pi}{2}\] , then write the value of \[\sqrt{\frac{1 + \cos 2x}{2}}\]

 

 

Q 4 | Page 42

If \[\frac{\pi}{2} < x < \pi,\] the write the value of \[\sqrt{2 + \sqrt{2 + 2 \cos 2x}}\] in the simplest form.

 
 
Q 5 | Page 42

If  \[\frac{\pi}{2} < x < \pi\], then write the value of \[\frac{\sqrt{1 - \cos 2x}}{1 + \cos 2x}\] .

 

 

Q 6 | Page 42

If \[\pi < x < \frac{3\pi}{2}\], then write the value of \[\sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}\] . 

 
Q 7 | Page 42

In a right angled triangle ABC, write the value of sin2 A + Sin2 B + Sin2 C.

 
Q 8 | Page 42

Write the value of \[\cos^2 76°  + \cos^2 16°  - \cos 76° \cos 16°\] 

 
Q 9 | Page 42

If \[\frac{\pi}{4} < x < \frac{\pi}{2}\], then write the value of \[\sqrt{1 - \sin 2x}\] .

 

 

Q 10 | Page 42

Write the value of \[\cos\frac{\pi}{7} \cos\frac{2\pi}{7} \cos\frac{4\pi}{7} .\]

  
Q 11 | Page 42

If \[\text{ tan } A = \frac{1 - \text{ cos } B}{\text{ sin } B}\]

, then find the value of tan2A.

 

 

Q 12 | Page 42

If  \[\text{ sin } x + \text{ cos } x = a\], then find the value of

\[\sin^6 x + \cos^6 x\] .
 

 

Q 13 | Page 42

If  \[\text{ sin } x + \text{ cos } x = a\], find the value of \[\left|\text { sin } x - \text{ cos } x \right|\] .

 

 

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle solutions [Pages 43 - 45]

Q 1 | Page 43
\[8 \sin\frac{x}{8} \cos \frac{x}{2}\cos\frac{x}{4} \cos\frac{x}{8}\]  is equal to 

 

  • 8 cos x

  • cos x

  •  8 sin x

  • sin x

Q 2 | Page 43
\[\frac{\sec 8A - 1}{\sec 4A - 1} =\]

 

  • \[\frac{\tan 2A}{\tan 8A}\]

     

  • \[\frac{\tan 8A}{\tan 2A}\]

     

  • \[\frac{\cot 8A}{\cot 2A}\]

     

  • none of these.

     
Q 3 | Page 43

The value of \[\cos \frac{\pi}{65} \cos \frac{2\pi}{65} \cos \frac{4\pi}{65} \cos \frac{8\pi}{65} \cos \frac{16\pi}{65} \cos \frac{32\pi}{65}\]  is 

  
  • \[\frac{1}{8}\]

     

  • \[\frac{1}{16}\]

     

  • \[\frac{1}{32}\]

     

  •  none of these

Q 4 | Page 43

If \[\cos 2x + 2 \cos x = 1\]  then, \[\left( 2 - \cos^2 x \right) \sin^2 x\]  is equal to 

 
 
  • 1

  • -1

  • \[- \sqrt{5}\]

     

  • \[\sqrt{5}\]

     

Q 5 | Page 43

For all real values of x, \[\cot x - 2 \cot 2x\] is equal to 

 
  • \[\tan 2x\]

     

  • \[\tan x\]

     

  • \[- \cot 3x\]

     

  • none of these

Q 6 | Page 43

The value of  \[2 \tan \frac{\pi}{10} + 3 \sec \frac{\pi}{10} - 4 \cos \frac{\pi}{10}\] is 

 
  • 0

  • \[\sqrt{5}\]

     

  • 1

  • none of these

Q 7 | Page 43

If in a  \[∆ ABC, \tan A + \tan B + \tan C = 0\], then

\[\cot A \cot B \cot C =\]
 

 

  • 6

  • 1

  • \[\frac{1}{6}\]

     

  •  none of these

Q 8 | Page 43

If \[\cos x = \frac{1}{2} \left( a + \frac{1}{a} \right),\]  and \[\cos 3 x = \lambda \left( a^3 + \frac{1}{a^3} \right)\] then \[\lambda =\]

 

 

  • \[\frac{1}{4}\]

     

  • \[\frac{1}{2}\]

     

  • 1

  • none of these

Q 9 | Page 43

If  \[2 \tan \alpha = 3 \tan \beta, \text{ then }  \tan \left( \alpha - \beta \right) =\]

 

  • \[\frac{\sin 2 \beta}{5 - \cos 2 \beta}\]

  • \[\frac{\cos 2 \beta}{5 - \cos 2 \beta}\]

     

  • \[\frac{\sin 2 \beta}{5 + \cos 2 \beta}\]

  •  none of these

Q 10 | Page 43

If \[\tan \alpha = \frac{1 - \cos \beta}{\sin \beta}\] , then

 
  • \[\tan 3  \alpha = \tan 2 \beta\]

  • \[\tan 2 \alpha = \tan \beta\]

     

  • \[\tan 2 \alpha = \tan \alpha\]

     

  • none of these 

Q 11 | Page 43

If \[\sin \alpha + \sin \beta = a \text{ and }  \cos \alpha - \cos \beta = b \text{ then }  \tan \frac{\alpha - \beta}{2} =\]

 

  • \[- \frac{a}{b}\]

     

  • \[- \frac{b}{a}\]

     

  • \[\sqrt{a^2 + b^2}\]

     

  • none of these

Q 12 | Page 43

The value of \[\left( \cot \frac{x}{2} - \tan \frac{x}{2} \right)^2 \left( 1 - 2 \tan x \cot 2 x \right)\] is 

 
  • 1

  • 2

  • 3

  • 4

Q 13 | Page 43

The value of \[\tan x \sin \left( \frac{\pi}{2} + x \right) \cos \left( \frac{\pi}{2} - x \right)\]

 
  • 1

  • -1

  • \[\frac{1}{2} \sin 2x\]

     

  • none of these.

Q 14 | Page 44

\[\sin^2 \left( \frac{\pi}{18} \right) + \sin^2 \left( \frac{\pi}{9} \right) + \sin^2 \left( \frac{7\pi}{18} \right) + \sin^2 \left( \frac{4\pi}{9} \right) =\]

  • 1

  • 2

  • 4

  • none of these. 

Q 15 | Page 44

If  \[5 \sin \alpha = 3 \sin \left( \alpha + 2 \beta \right) \neq 0\] , then \[\tan \left( \alpha + \beta \right)\]  is equal to

 
  • \[2 \tan \beta\]

  • \[3 \tan \beta\]

  • \[4 \tan \beta\]

  • \[6 \tan \beta\]

Q 16 | Page 44

\[2 \text{ cos } x - \ cos  3x - \cos 5x - 16 \cos^3 x \sin^2 x\]

  • 2

  • 1

  • 0

  • -1

Q 17 | Page 44

If \[A = 2 \sin^2 x - \cos 2x\] , then A lies in the interval

  • \[\left[ - 1, 3 \right]\]

  • \[\left[ 1, 2 \right]\] 

  • \[\left[ - 2, 4 \right]\]

  •  none of these 

Q 18 | Page 44

The value of \[\frac{\cos 3x}{2 \cos 2x - 1}\]  is equal to

   
  •  cos x

  • sin x

  • tan x

  • none of these

Q 19 | Page 44

If \[\tan \left( \pi/4 + x \right) + \tan \left( \pi/4 - x \right) = \lambda \sec 2x, \text{ then } \]

  • 3

  • 4

  • 1

  • 2

Q 20 | Page 44

The value of  \[\cos^2 \left( \frac{\pi}{6} + x \right) - \sin^2 \left( \frac{\pi}{6} - x \right)\] is 

  
  • \[\frac{1}{2} \cos 2x\]

  • 0

  • \[- \frac{1}{2} \cos 2x\]

  • \[\frac{1}{2}\]

Q 21 | Page 44

\[\frac{\sin 3x}{1 + 2 \cos 2x}\]   is equal to

  • cos x

  • sin x

  •  – cos x

  • sin x

Q 22 | Page 44

The value of  \[2 \sin^2 B + 4 \cos \left( A + B \right) \sin A \sin B + \cos 2 \left( A + B \right)\] is 

  • 0

  •  cos 3A

  • cos 2A

  •  none of these

Q 23 | Page 44

The value of \[\frac{2\left( \sin 2x + 2 \cos^2 x - 1 \right)}{\cos x - \sin x - \cos 3x + \sin 3x}\] is 

 
  •  cos x

  • sec x

  •  cosec x

  • sin x

Q 24 | Page 44

\[2 \left( 1 - 2 \sin^2 7x \right) \sin 3x\]  is equal to

  • \[\sin 17x - \sin 11x\]

  • \[\sin 11x - \sin 17x\]

  • \[\cos 17x - \cos 11x\]

  • \[\cos 17x + \cos 11x\]

Q 25 | Page 44

If α and β are acute angles satisfying \[\cos 2 \alpha = \frac{3 \cos 2 \beta - 1}{3 - \cos 2 \beta}\] , then tan α =

 
  • \[\sqrt{2} \tan \beta\]

  • \[\frac{1}{\sqrt{2}}\tan \beta\]

  • \[\sqrt{2} \cot \beta\]

  • \[\frac{1}{\sqrt{2}} \cot \beta\]

Q 26 | Page 44

If  \[\tan \frac{x}{2} = \frac{\sqrt{1 - e}}{1 + e} \tan \frac{\alpha}{2}\] , then \[\cos \alpha =\]

  • \[1 - e \cos \left( \cos x + e \right)\]

  • \[\frac{1 + e \cos x}{\cos x - e}\]

  • \[\frac{1 - e \cos x}{\cos x - e}\]

  • \[\frac{\cos x - e}{1 - e \cos x}\]

Q 27 | Page 45

If  \[\left( 2^n + 1 \right) x = \pi,\] then \[2^n \cos x \cos 2x \cos 2^2 x . . . \cos 2^{n - 1} x = 1\]

 

  • -1

  • 1

  • 1/2

  • None of these

Q 28 | Page 45

If \[\tan x = t\] then \[\tan 2x + \sec 2x =\]

 

  • \[\frac{1 + t}{1 - t}\]

     

  • \[\frac{1 - t}{1 + t}\]

     

  • \[\frac{2t}{1 - t}\]

     

  • \[\frac{2t}{1 + t}\]

     

Q 29 | Page 45

The value of \[\cos^4 x + \sin^4 x - 6 \cos^2 x \sin^2 x\] is 

  • cos 2x

  •  sin 2x

  • cos 4x

  • none of these

Q 30 | Page 45

The value of \[\cos \left( 36°  - A \right) \cos \left( 36° + A \right) + \cos \left( 54°  - A \right) \cos \left( 54°  + A \right)\] is 

 
  • cos 2A

  • sin 2A

  • cos A

  • 0

Q 31 | Page 45

The value of \[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right)\] is

 
  •  cot 3x

  • 2cot 3x

  •  tan 3x

  • 3 tan 3x

Q 32 | Page 45

The value of \[\tan x + \tan \left( \frac{\pi}{3} + x \right) + \tan \left( \frac{2\pi}{3} + x \right)\] is 

 
  • 3 tan 3x

  • tan 3x

  • 3 cot 3x

  •  cot 3x

Q 33 | Page 45

The value of \[\frac{\sin 5 \alpha - \sin 3\alpha}{\cos 5 \alpha + 2 \cos 4\alpha + \cos 3\alpha} =\]

 
  • \[\cot \alpha/2\]

     

  • \[\cot \alpha\]

     

  • \[\tan \alpha/2\]

     

  • None of these 

Q 34 | Page 45
\[\frac{\sin 5x}{\sin x}\]  is equal to

 

  • \[16 \cos^4 x - 12 \cos^2 x + 1\]

     

  • \[16 \cos^4 x + 12 \cos^2 x + 1\]

     

  • \[16 \cos^4 x - 12 \cos^2 x - 1\]

     

  • \[16 \cos^4 x + 12 \cos^2 x - 1\]

     

Q 35 | Page 45

If \[n = 1, 2, 3, . . . , \text{ then }  \cos \alpha \cos 2 \alpha \cos 4 \alpha . . . \cos 2^{n - 1} \alpha\] is equal to

 

  • \[\frac{\sin 2n \alpha}{2n \sin \alpha}\]

  • \[\frac{\sin 2^n \alpha}{2^n \sin 2^{n - 1} \alpha}\]

     

  • \[\frac{\sin 4^{n - 1} \alpha}{4^{n - 1} \sin \alpha}\]

  • \[\frac{\sin 2^n \alpha}{2^n \sin \alpha}\]

     

Q 36 | Page 45

If \[\text{ tan } x = \frac{a}{b}\], then \[b \cos 2x + a \sin 2x\]

 

 

  • a

  • b

  • \[\frac{a}{b}\]

     

  • \[\frac{b}{a}\]

     

Q 37 | Page 45

If \[\tan\alpha = \frac{1}{7}, \tan\beta = \frac{1}{3}\], then

\[\cos2\alpha\]   is equal to

 
  • \[\sin2\beta\]

  • \[\sin4\beta\]

     

  • \[\sin3\beta\]

     

  • \[\cos2\beta\]

     

Q 38 | Page 45

The value of\[\cos^2 48°- \sin^2 12°\]  is 

 
  • \[\frac{\sqrt{5} + 1}{8}\]

     

  • \[\frac{\sqrt{5} - 1}{8}\]

     

  • \[\frac{\sqrt{5} + 1}{5}\]

     

  • \[\frac{\sqrt{5} + 1}{2\sqrt{2}}\]

     

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Ex. 9.1Ex. 9.2Ex. 9.3Others

RD Sharma Mathematics Class 11

Mathematics Class 11

RD Sharma solutions for Class 11 Mathematics chapter 9 - Values of Trigonometric function at multiples and submultiples of an angle

RD Sharma solutions for Class 11 Maths chapter 9 (Values of Trigonometric function at multiples and submultiples of an angle) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 9 Values of Trigonometric function at multiples and submultiples of an angle are Sine and Cosine Formulae and Their Applications, Values of Trigonometric Functions at Multiples and Submultiples of an Angle, Transformation Formulae, Graphs of Trigonometric Functions, Conversion from One Measure to Another, 90 Degree Plusminus X Function, Negative Function Or Trigonometric Functions of Negative Angles, Truth of the Identity, Trigonometric Equations, Trigonometric Functions of Sum and Difference of Two Angles, Domain and Range of Trigonometric Functions, Signs of Trigonometric Functions, Introduction of Trigonometric Functions, Concept of Angle, Expressing Sin (X±Y) and Cos (X±Y) in Terms of Sinx, Siny, Cosx and Cosy and Their Simple Applications, 3X Function, 2X Function, 180 Degree Plusminus X Function.

Using RD Sharma Class 11 solutions Values of Trigonometric function at multiples and submultiples of an angle exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

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