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RD Sharma solutions for Class 11 Mathematics chapter 8 - Transformation formulae

Mathematics Class 11

RD Sharma Mathematics Class 11 Chapter 8: Transformation formulae

Ex. 8.10Ex. 8.20Others

Chapter 8: Transformation formulae Exercise 8.10 solutions [Pages 6 - 7]

Ex. 8.10 | Q 1.1 | Page 6

Express the following as the sum or difference of sines and cosines:
2 sin 3x cos x

Ex. 8.10 | Q 1.2 | Page 6

Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa

Ex. 8.10 | Q 1.3 | Page 6

Express the following as the sum or difference of sines and cosines:
2 sin 4x sin 3x

Ex. 8.10 | Q 1.4 | Page 6

Express the following as the sum or difference of sines and cosines:
2 cos 7x cos 3x

Ex. 8.10 | Q 2.1 | Page 6

Prove that:

$2\sin\frac{5\pi}{12}\sin\frac{\pi}{12} = \frac{1}{2}$

Ex. 8.10 | Q 2.2 | Page 6

Prove that:

$2\cos\frac{5\pi}{12}\cos\frac{\pi}{12} = \frac{1}{2}$
Ex. 8.10 | Q 2.3 | Page 6

Prove that:

$2\sin\frac{5\pi}{12}\cos\frac{\pi}{12} = \frac{\sqrt{3} + 2}{2}$
Ex. 8.10 | Q 3.1 | Page 6

Show that :

$\sin 50^\circ \cos 85^\circ = \frac{1 - \sqrt{2} \sin 35^\circ}{2\sqrt{2}}$
Ex. 8.10 | Q 3.2 | Page 6

Show that :

$\sin 25^\circ \cos 115^\circ = \frac{1}{2}\left( \sin 140^\circ - 1 \right)$
Ex. 8.10 | Q 4 | Page 7
$\text{ Prove that }4 \cos x \cos\left( \frac{\pi}{3} + x \right) \cos \left( \frac{\pi}{3} - x \right) = \cos 3x .$

Ex. 8.10 | Q 5.1 | Page 7

Prove that:
cos 10° cos 30° cos 50° cos 70° = $\frac{3}{16}$

Ex. 8.10 | Q 5.2 | Page 7

Prove that:
cos 40° cos 80° cos 160° = $- \frac{1}{8}$

Ex. 8.10 | Q 5.3 | Page 7

Prove that:
sin 20° sin 40° sin 80° = $\frac{\sqrt{3}}{8}$

Ex. 8.10 | Q 5.4 | Page 7

Prove that:
cos 20° cos 40° cos 80° = $\frac{1}{8}$

Ex. 8.10 | Q 5.5 | Page 7

Prove that:
tan 20° tan 40° tan 60° tan 80° = 3

Ex. 8.10 | Q 5.6 | Page 7

Prove that:
tan 20° tan 30° tan 40° tan 80° = 1

Ex. 8.10 | Q 5.7 | Page 7

Prove that:
sin 10° sin 50° sin 60° sin 70° = $\frac{\sqrt{3}}{16}$

Ex. 8.10 | Q 5.8 | Page 7

Prove that:
sin 20° sin 40° sin 60° sin 80° = $\frac{3}{16}$

Ex. 8.10 | Q 6.1 | Page 7

Show that:
sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 0

Ex. 8.10 | Q 6.2 | Page 7

Show that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0

Ex. 8.10 | Q 7 | Page 7

Prove that
$\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x$

Ex. 8.10 | Q 8 | Page 7

If α + β = $\frac{\pi}{2}$, show that the maximum value of cos α cos β is $\frac{1}{2}$.

Chapter 8: Transformation formulae Exercise 8.20 solutions [Pages 17 - 19]

Ex. 8.20 | Q 1.1 | Page 17

Express each of the following as the product of sines and cosines:
sin 12x + sin 4x

Ex. 8.20 | Q 1.2 | Page 17

Express each of the following as the product of sines and cosines:
sin 5x − sin x

Ex. 8.20 | Q 1.3 | Page 17

Express each of the following as the product of sines and cosines:
cos 12x + cos 8x

Ex. 8.20 | Q 1.4 | Page 17

Express each of the following as the product of sines and cosines:
cos 12x - cos 4x

Ex. 8.20 | Q 1.5 | Page 17

Express each of the following as the product of sines and cosines:
sin 2x + cos 4x

Ex. 8.20 | Q 2.1 | Page 17

Prove that:
sin 38° + sin 22° = sin 82°

Ex. 8.20 | Q 2.2 | Page 17

Prove that:
cos 100° + cos 20° = cos 40°

Ex. 8.20 | Q 2.3 | Page 17

Prove that:
sin 50° + sin 10° = cos 20°

Ex. 8.20 | Q 2.4 | Page 17

Prove that:
sin 23° + sin 37° = cos 7°

Ex. 8.20 | Q 2.5 | Page 17

Prove that:
sin 105° + cos 105° = cos 45°

Ex. 8.20 | Q 2.6 | Page 17

Prove that:
sin 40° + sin 20° = cos 10°

Ex. 8.20 | Q 3.1 | Page 17

Prove that:
cos 55° + cos 65° + cos 175° = 0

Ex. 8.20 | Q 3.2 | Page 17

Prove that:
sin 50° − sin 70° + sin 10° = 0

Ex. 8.20 | Q 3.3 | Page 17

Prove that:
cos 80° + cos 40° − cos 20° = 0

Ex. 8.20 | Q 3.4 | Page 17

Prove that:
cos 20° + cos 100° + cos 140° = 0

Ex. 8.20 | Q 3.5 | Page 17

Prove that:
$\sin\frac{5\pi}{18} - \cos\frac{4\pi}{9} = \sqrt{3} \sin\frac{\pi}{9}$

Ex. 8.20 | Q 3.6 | Page 17

Prove that:

$\cos\frac{\pi}{12} - \sin\frac{\pi}{12} = \frac{1}{\sqrt{2}}$

Ex. 8.20 | Q 3.7 | Page 17

Prove that:

sin 80° − cos 70° = cos 50°
Ex. 8.20 | Q 3.8 | Page 17

Prove that:

sin 51° + cos 81° = cos 21°
Ex. 8.20 | Q 4.1 | Page 18

Prove that:

$\cos\left( \frac{3\pi}{4} + x \right) - \cos\left( \frac{3\pi}{4} - x \right) = - \sqrt{2} \sin x$

Ex. 8.20 | Q 4.2 | Page 18

Prove that:

$\cos\left( \frac{\pi}{4} + x \right) + \cos\left( \frac{\pi}{4} - x \right) = \sqrt{2} \cos x$

Ex. 8.20 | Q 5.1 | Page 18

Prove that:

$\sin 65^\circ + \cos 65^\circ = \sqrt{2} \cos 20^\circ$
Ex. 8.20 | Q 5.2 | Page 18

Prove that:
sin 47° + cos 77° = cos 17°

Ex. 8.20 | Q 6.1 | Page 18

Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A

Ex. 8.20 | Q 6.2 | Page 18

Prove that:
cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A

Ex. 8.20 | Q 6.3 | Page 18

Prove that:
sin A + sin 2A + sin 4A + sin 5A = 4 cos $\frac{A}{2}$$\frac{3A}{2}$

Ex. 8.20 | Q 6.4 | Page 18

Prove that:
sin 3A + sin 2A − sin A = 4 sin A cos $\frac{A}{2}$ $\frac{3A}{2}$

Ex. 8.20 | Q 6.5 | Page 18

Prove that:

cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −$\frac{3}{4}$

Ex. 8.20 | Q 6.6 | Page 18
Prove that:
$\sin\frac{x}{2}\sin\frac{7x}{2} + \sin\frac{3x}{2}\sin\frac{11x}{2} = \sin 2x \sin 5x .$

Ex. 8.20 | Q 6.7 | Page 18
Prove that:
$\cos x \cos \frac{x}{2} - \cos 3x \cos\frac{9x}{2} = \sin 7x \sin 8x$

Ex. 8.20 | Q 7.1 | Page 18

Prove that:

$\frac{\sin A + \sin 3A}{\cos A - \cos 3A} = \cot A$

Ex. 8.20 | Q 7.2 | Page 18

Prove that:

$\frac{\sin 9A - \sin 7A}{\cos 7A - \cos 9A} = \cot 8A$
Ex. 8.20 | Q 7.3 | Page 18

Prove that:

$\frac{\sin A - \sin B}{\cos A + \cos B} = \tan\frac{A - B}{2}$
Ex. 8.20 | Q 7.4 | Page 18

Prove that:

$\frac{\sin A + \sin B}{\sin A - \sin B} = \tan \left( \frac{A + B}{2} \right) \cot \left( \frac{A - B}{2} \right)$
Ex. 8.20 | Q 7.5 | Page 18

Prove that:

$\frac{\cos A + \cos B}{\cos B - \cos A} = \cot \left( \frac{A + B}{2} \right) \cot \left( \frac{A - B}{2} \right)$
Ex. 8.20 | Q 8.01 | Page 18

Prove that:

$\frac{\sin A + \sin 3A + \sin 5A}{\cos A + \cos 3A + \cos 5A} = \tan 3A$

Ex. 8.20 | Q 8.02 | Page 18

Prove that:

$\frac{\cos 3A + 2 \cos 5A + \cos 7A}{\cos A + 2 \cos 3A + \cos 5A} = \frac{\cos 5A}{\cos 3A}$
Ex. 8.20 | Q 8.03 | Page 18

Prove that:

$\frac{\cos 4A + \cos 3A + \cos 2A}{\sin 4A + \sin 3A + \sin 2A} = \cot 3A$

Ex. 8.20 | Q 8.04 | Page 18

Prove that:

$\frac{\sin 3A + \sin 5A + \sin 7A + \sin 9A}{\cos 3A + \cos 5A + \cos 7A + \cos 9A} = \tan 6A$
Ex. 8.20 | Q 8.05 | Page 18

Prove that:

$\frac{\sin 5A - \sin 7A + \sin 8A - \sin 4A}{\cos 4A + \cos 7A - \cos 5A - \cos 8A} = \cot 6A$
Ex. 8.20 | Q 8.06 | Page 18

Prove that:

$\frac{\sin 5A \cos 2A - \sin 6A \cos A}{\sin A \sin 2A - \cos 2A \cos 3A} = \tan A$
Ex. 8.20 | Q 8.07 | Page 18

Prove that:

$\frac{\sin 11A \sin A + \sin 7A \sin 3A}{\cos 11A \sin A + \cos 7A \sin 3A} = \tan 8A$
Ex. 8.20 | Q 8.08 | Page 18

Prove that:

$\frac{\sin 3A \cos 4A - \sin A \cos 2A}{\sin 4A \sin A + \cos 6A \cos A} = \tan 2A$
Ex. 8.20 | Q 8.09 | Page 18

Prove that:

$\frac{\sin A \sin 2A + \sin 3A \sin 6A}{\sin A \cos 2A + \sin 3A \cos 6A} = \tan 5A$
Ex. 8.20 | Q 8.1 | Page 18

Prove that:

$\frac{\sin A + 2 \sin 3A + \sin 5A}{\sin 3A + 2 \sin 5A + \sin 7A} = \frac{\sin 3A}{\sin 5A}$
Ex. 8.20 | Q 8.11 | Page 18

Prove that:

$\frac{\sin \left( \theta + \phi \right) - 2 \sin \theta + \sin \left( \theta - \phi \right)}{\cos \left( \theta + \phi \right) - 2 \cos \theta + \cos \left( \theta - \phi \right)} = \tan \theta$
Ex. 8.20 | Q 9.1 | Page 19

Prove that:

$\sin \alpha + \sin \beta + \sin \gamma - \sin (\alpha + \beta + \gamma) = 4 \sin \left( \frac{\alpha + \beta}{2} \right) \sin \left( \frac{\beta + \gamma}{2} \right) \sin \left( \frac{\gamma + \alpha}{2} \right)$

Ex. 8.20 | Q 9.2 | Page 19

Prove that:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C

Ex. 8.20 | Q 10 | Page 19
$\text{ If } \cos A + \cos B = \frac{1}{2}\text{ and }\sin A + \sin B = \frac{1}{4},\text{ prove that }\tan\left( \frac{A + B}{2} \right) = \frac{1}{2} .$

Ex. 8.20 | Q 11 | Page 19

If cosec A + sec A = cosec B + sec B, prove that tan A tan B = $\cot\frac{A + B}{2}$.

Ex. 8.20 | Q 12 | Page 19
$\text{ If }\sin 2A = \lambda \sin 2B, \text{ prove that }\frac{\tan (A + B)}{\tan (A - B)} = \frac{\lambda + 1}{\lambda - 1}$

Ex. 8.20 | Q 13.1 | Page 19

Prove that:

$\frac{\cos (A + B + C) + \cos ( - A + B + C) + \cos (A - B + C) + \cos (A + B - C)}{\sin (A + B + C) + \sin ( - A + B + C) + \sin (A - B + C) - \sin (A + B - C)} = \cot C$
Ex. 8.20 | Q 13.2 | Page 19

Prove that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0

Ex. 8.20 | Q 14 | Page 19
$\text{ If }\frac{\cos (A - B)}{\cos (A + B)} + \frac{\cos (C + D)}{\cos (C - D)} = 0, \text {Prove that }\tan A \tan B \tan C \tan D = - 1$

Ex. 8.20 | Q 15 | Page 19

If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ

Ex. 8.20 | Q 16 | Page 19

If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.

Ex. 8.20 | Q 17 | Page 19

If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), prove that tan A tan B tan C + tan D = 0.

Ex. 8.20 | Q 18 | Page 19

If $x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)$, prove that $xy + yz + zx = 0$

Ex. 8.20 | Q 19 | Page 19

If $m \sin\theta = n \sin\left( \theta + 2\alpha \right)$, prove that $\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}$

Chapter 8: Transformation formulae solutions [Pages 20 - 21]

Q 1 | Page 20

If (cos α + cos β)2 + (sin α + sin β)2 = $\lambda \cos^2 \left( \frac{\alpha - \beta}{2} \right)$, write the value of λ.

Q 2 | Page 20

Write the value of sin $\frac{\pi}{12}$ sin $\frac{5\pi}{12}$.

Q 3 | Page 20

If sin A + sin B = α and cos A + cos B = β, then write the value of tan $\left( \frac{A + B}{2} \right)$.

Q 4 | Page 20

If cos A = m cos B, then write the value of $\cot\frac{A + B}{2} \cot\frac{A - B}{2}$.

Q 5 | Page 20

Write the value of the expression $\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}$

Q 6 | Page 20

If A + B = $\frac{\pi}{3}$ and cos A + cos B = 1, then find the value of cos $\frac{A - B}{2}$.

Q 7 | Page 20

Write the value of $\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}$

Q 8 | Page 21

If sin 2A = λ sin 2B, then write the value of $\frac{\lambda + 1}{\lambda - 1}$

Q 9 | Page 21

Write the value of $\frac{\sin A + \sin 3A}{\cos A + \cos 3A}$

Q 10 | Page 21

If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.

Chapter 8: Transformation formulae solutions [Pages 21 - 22]

Q 1 | Page 21

cos 40° + cos 80° + cos 160° + cos 240° =

• 0

• 1

• $\frac{1}{2}$

• $- \frac{1}{2}$

Q 2 | Page 21

sin 163° cos 347° + sin 73° sin 167° =

• 0

• $\frac{1}{2}$

• 1

• None of these

Q 3 | Page 21

If sin 2 θ + sin 2 ϕ = $\frac{1}{2}$ and cos 2 θ + cos 2 ϕ = $\frac{3}{2}$, then cos2 (θ − ϕ) =

• $\frac{3}{8}$

• $\frac{5}{8}$

• $\frac{3}{4}$

• $\frac{5}{4}$

Q 4 | Page 21

The value of cos 52° + cos 68° + cos 172° is

• 0

• 1

• 2

• 3/2

Q 5 | Page 21

The value of sin 78° − sin 66° − sin 42° + sin 60° is

• $\frac{1}{2}$

• $- \frac{1}{2}$

• −1

• None of these

Q 6 | Page 21

If sin α + sin β = a and cos α − cos β = b, then tan $\frac{\alpha - \beta}{2}$=

• $- \frac{a}{b}$

• $- \frac{b}{a}$

• $\sqrt{a^2 + b^2}$

•  None of these

Q 7 | Page 21

cos 35° + cos 85° + cos 155° =

• 0

• $\frac{1}{\sqrt{3}}$

• $\frac{1}{\sqrt{2}}$

•  cos 275°

Q 8 | Page 21

The value of sin 50° − sin 70° + sin 10° is equal to

• 1

• 0

• 1/2

• 2

Q 9 | Page 21

sin 47° + sin 61° − sin 11° − sin 25° is equal to

• sin 36°

• cos 36°

• sin 7°

• cos 7°

Q 10 | Page 21

If cos A = m cos B, then $\cot\frac{A + B}{2} \cot\frac{B - A}{2}$=

• $\frac{m - 1}{m + 1}$

• $\frac{m + 2}{m - 2}$

• $\frac{m + 1}{m - 1}$

•  None of these

Q 11 | Page 21

If A, B, C are in A.P., then $\frac{\sin A - \sin C}{\cos C - \cos A}$=

•  tan B

• cot B

• tan 2 B

• None of these

Q 12 | Page 22

If sin (B + C − A), sin (C + A − B), sin (A + B − C) are in A.P., then cot A, cot B and cot Care in

• GP

• HP

• AP

• None of these

Q 13 | Page 22

If sin x + sin y = $\sqrt{3}$ (cos y − cos x), then sin 3x + sin 3y =

• 2 sin 3x

• 0

• 1

• none of these

Q 14 | Page 22

If $\tan\alpha = \frac{x}{x + 1}$ and

$\tan\beta = \frac{1}{2x + 1}$, then
$\tan\beta = \frac{1}{2x + 1}$ is equal to

• $\frac{\pi}{2}$

• $\frac{\pi}{2}$

• $\frac{\pi}{2}$

• $\frac{\pi}{2}$

Chapter 8: Transformation formulae

Ex. 8.10Ex. 8.20Others

RD Sharma Mathematics Class 11 RD Sharma solutions for Class 11 Mathematics chapter 8 - Transformation formulae

RD Sharma solutions for Class 11 Maths chapter 8 (Transformation formulae) 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.

Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 11 Mathematics chapter 8 Transformation formulae are Transformation Formulae, Values of Trigonometric Functions at Multiples and Submultiples of an Angle, Sine and Cosine Formulae and Their Applications, 180 Degree Plusminus X Function, 2X Function, 3X Function, Expressing Sin (X±Y) and Cos (X±Y) in Terms of Sinx, Siny, Cosx and Cosy and Their Simple Applications, Concept of Angle, Introduction of Trigonometric Functions, Signs of Trigonometric Functions, Domain and Range of Trigonometric Functions, Trigonometric Functions of Sum and Difference of Two Angles, Trigonometric Equations, Truth of the Identity, Negative Function Or Trigonometric Functions of Negative Angles, 90 Degree Plusminus X Function, Conversion from One Measure to Another, Graphs of Trigonometric Functions.

Using RD Sharma Class 11 solutions Transformation formulae 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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