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RD Sharma solutions for Class 11 Mathematics chapter 5 - Trigonometric Functions

Chapter 5 : Trigonometric Functions

Pages 0 - 19

Prove the following identites

sec4x - sec2x = tan4x + tan2x

Prove the following identities
$\sin^6 x + \cos^6 x = 1 - 3 \sin^2 x \cos^2 x$

Prove the following identities
$\left( cosec x - \sin x \right) \left( \sec x - \cos x \right) \left( \tan x + \cot x \right) = 1$

Prove the following identities
$cosec x \left( \sec x - 1 \right) - \cot x \left( 1 - \cos x \right) = \tan x - \sin x$

Prove the following identities
$\frac{1 - \sin x \cos x}{\cos x \left( \sec x - cosec x \right)} \cdot \frac{\sin^2 x - \cos^2 x}{\sin^3 x + \cos^3 x} = \sin x$

Prove the following identitie

$\frac{\tan x}{1 - \cot x} + \frac{\cot x}{1 - \tan x} = \left( \sec x cossec x + 1 \right)$

Prove the following identities
$\frac{\sin^3 x + \cos^3 x}{\sin x + \cos x} + \frac{\sin^3 x - \cos^3 x}{\sin x - \cos x} = 2$

Prove the following identities
$\left( \sec x \sec y + \tan x \tan y \right)^2 - \left( \sec x \tan y + \tan x \sec y \right)^2 = 1$

Prove the following identities
$\frac{\cos x}{1 - \sin x} = \frac{1 + \cos x + \sin x}{1 + \cos x - \sin x}$

Prove the following identities

$\frac{\tan^3 x}{1 + \tan^2 x} + \frac{\cot^3 x}{1 + \cot^2 x} = \frac{1 - 2 \sin^2 x \cos^2 x}{\sin x \cos x}$

Prove the following identities
$1 - \frac{\sin^2 x}{1 + \cot x} - \frac{\cos^2 x}{1 + \tan x} = \sin x \cos x$

Prove the following identities

$\left( \frac{1}{\sec^2 x - \cos^2 x} + \frac{1}{{cosec}^2 x - \sin^2 x} \right) \sin^2 x \cos^2 x = \frac{1 - \sin^2 x \cos^2 x}{2 + \sin^2 x \cos^2 x}$

Prove the following identities
$\left( 1 + \tan \alpha \tan \beta \right)^2 + \left( \tan \alpha - \tan \beta \right)^2 = \sec^2 \alpha \sec^2 \beta$

Prove the following identities

$\frac{\left( 1 + \cot x + \tan x \right) \left( \sin x - \cos x \right)}{\sec^3 x - {cosec}^3 x} = \sin^2 x \cos^2 x$

Prove the following identities

$\frac{2 \sin x \cos x - \cos x}{1 - \sin x + \sin^2 x - \cos^2 x} = \cot x$

Prove the following identities

$\cos x \left( \tan x + 2 \right) \left( 2 \tan x + 1 \right) = 2 \sec x + 5 \sin x$

If $x = \frac{2 \sin x}{1 + \cos x + \sin x}$, then prove that

$\frac{1 - \cos x + \sin x}{1 + \sin x}$ is also equal to a.

If $\sin x = \frac{a^2 - b^2}{a^2 + b^2}$, then the values of tan x, sec x and cosec x

If $\tan x = \frac{b}{a}$ , then find the values of $\sqrt{\frac{a + b}{a - b}} + \sqrt{\frac{a - b}{a + b}}$.

If $\tan x = \frac{a}{b},$ show that

$\frac{a \sin x - b \cos x}{a \sin x + b \cos x} = \frac{a^2 - b^2}{a^2 + b^2}$

If $cosec x - \sin x = a^3 , \sec x - \cos x = b^3$, then prove that $a^2 b^2 \left( a^2 + b^2 \right) = 1$

If $\cot x \left( 1 + \sin x \right) = 4 m \text{ and }\cot x \left( 1 - \sin x \right) = 4 n,$ $\left( m^2 + n^2 \right)^2 = mn$

If $\sin x + \cos x = m$, then prove that $\sin^6 x + \cos^6 x = \frac{4 - 3 \left( m^2 - 1 \right)^2}{4}$, where $m^2 \leq 2$

If $a = \sec x - \tan x \text{ and }b = cosec x + \cot x$, then shown that  $ab + a - b + 1 = 0$

Prove the:
$\sqrt{\frac{1 - \sin x}{1 + \sin x}} + \sqrt{\frac{1 + \sin x}{1 - \sin x}} = - \frac{2}{\cos x},\text{ where }\frac{\pi}{2} < x < \pi$

If $T_n = \sin^n x + \cos^n x$, prove that $\frac{T_3 - T_5}{T_1} = \frac{T_5 - T_7}{T_3}$

If $T_n = \sin^n x + \cos^n x$, prove that  $2 T_6 - 3 T_4 + 1 = 0$

If $T_n = \sin^n x + \cos^n x$, prove that $6 T_{10} - 15 T_8 + 10 T_6 - 1 = 0$

Page 25

Q 1.1 | Page 25

Find the value of the other five trigonometric functions

$\cot x = \frac{12}{5},$ x in quadrant III
Q 1.2 | Page 25

Find the value of the other five trigonometric functions

$\cos x = - \frac{1}{2},$ x in quadrant II
Q 1.3 | Page 25

Find the value of the other five trigonometric functions
$\tan x = \frac{3}{4},$ x in quadrant III

Q 1.4 | Page 25

Find the value of the other five trigonometric functions
$\sin x = \frac{3}{5},$ x in quadrant I

Q 2 | Page 25

If sin $x = \frac{12}{13}$ and x lies in the second quadrant, find the value of sec x + tan x.

Q 3 | Page 25

If sin$x = \frac{3}{5}, \tan y = \frac{1}{2}\text{ and }\frac{\pi}{2} < x < \pi < y < \frac{3\pi}{2},$  find the value of 8 tan $x - \sqrt{5} \sec y$

Q 4 | Page 25

If sin x + cos x = 0 and x lies in the fourth quadrant, find sin x and cos x.

Q 5 | Page 25

If $\cos x = - \frac{3}{5}\text{ and }\pi < x < \frac{3\pi}{2}$ find the values of other five trigonometric functions and hence evaluate $\frac{cosec x + \cot x}{\sec x - \tan x}$

Pages 39 - 40

Q 1.01 | Page 39

Find the value of the following trigonometric ratio:

$\sin\frac{5\pi}{3}$

Q 1.02 | Page 39

Find the value of the following trigonometric ratio:
sin 17π

Q 1.03 | Page 39

Find the value of the following trigonometric ratio:
$\tan\frac{11\pi}{6}$

Q 1.04 | Page 39

Find the value of the following trigonometric ratio:

$\cos\left( - \frac{25\pi}{4} \right)$
Q 1.05 | Page 39

Find the value of the following trigonometric ratio:
$\tan \frac{7\pi}{4}$

Q 1.06 | Page 39

Find the values of the following trigonometric ratio:

$\sin\frac{17\pi}{6}$

Q 1.07 | Page 39

Find the values of the following trigonometric ratio:

$\cos\frac{19\pi}{6}$

Q 1.08 | Page 39

Find the values of the following trigonometric ratio:

$\sin\left( - \frac{11\pi}{6} \right)$

Q 1.09 | Page 39

Find the values of the following trigonometric ratio:

$cosec\left( - \frac{20\pi}{3} \right)$

Q 1.1 | Page 39

Find the values of the following trigonometric ratio:

$\tan\left( - \frac{13\pi}{4} \right)$

Q 1.11 | Page 39

Find the values of the following trigonometric ratio:

$\cos\frac{19\pi}{4}$
Q 1.12 | Page 39

Find the values of the following trigonometric ratio:

$\sin\frac{41\pi}{4}$
Q 1.13 | Page 39

Find the values of the following trigonometric ratio:

$\cos\frac{39\pi}{4}$
Q 1.14 | Page 39

Find the values of the following trigonometric ratio:

$\sin\frac{151\pi}{6}$
Q 2.1 | Page 39

Prove that:  tan 225° cot 405° + tan 765° cot 675° = 0

Q 2.2 | Page 39

Prove that:

$\sin\frac{8\pi}{3}\cos\frac{23\pi}{6} + \cos\frac{13\pi}{3}\sin\frac{35\pi}{6} = \frac{1}{2}$

Q 2.3 | Page 39

Prove that: cos 24° + cos 55° + cos 125° + cos 204° + cos 300° = $\frac{1}{2}$

Q 2.4 | Page 39

Prove that: tan (−225°) cot (−405°) −tan (−765°) cot (675°) = 0

Q 2.5 | Page 39
Prove that:cos 570° sin 510° + sin (−330°) cos (−390°) = 0

Q 2.6 | Page 39

Prove that: $\tan\frac{11\pi}{3} - 2\sin\frac{4\pi}{6} - \frac{3}{4} {cosec}^2 \frac{\pi}{4} + 4 \cos^2 \frac{17\pi}{6} = \frac{3 - 4\sqrt{3}}{2}$

Q 2.7 | Page 39

Prove that:

$3\sin\frac{\pi}{6}\sec\frac{\pi}{3} - 4\sin\frac{5\pi}{6}\cot\frac{\pi}{4} = 1$

Q 3.1 | Page 39

Prove that:
$\frac{\cos (2\pi + x) cosec (2\pi + x) \tan (\pi/2 + x)}{\sec(\pi/2 + x)\cos x \cot(\pi + x)} = 1$

Q 3.2 | Page 39

Prove that

$\frac{cosec(90^\circ + x) + \cot(450^\circ + x)}{cosec(90^\circ - x) + \tan(180^\circ - x)} + \frac{\tan(180^\circ + x) + \sec(180^\circ - x)}{\tan(360^\circ + x) - \sec( - x)} = 2$

Q 3.3 | Page 39

Prove that

$\frac{\sin(180^\circ + x) \cos(90^\circ + x) \tan(270^\circ - x) \cot(360^\circ - x)}{\sin(360^\circ - x) \cos(360^\circ + x) cosec( - x) \sin(270^\circ + x)} = 1$

Q 3.4 | Page 39

Prove that

$\left\{ 1 + \cot x - \sec\left( \frac{\pi}{2} + x \right) \right\}\left\{ 1 + \cot x + \sec\left( \frac{\pi}{2} + x \right) \right\} = 2\cot x$

Q 3.5 | Page 39

Prove that

$\frac{\tan (90^\circ - x) \sec(180^\circ - x) \sin( - x)}{\sin(180^\circ + x) \cot(360^\circ - x) cosec(90^\circ - x)} = 1$

Q 4 | Page 40

Prove that:
$\sin^2 \frac{\pi}{18} + \sin^2 \frac{\pi}{9} + \sin^2 \frac{7\pi}{18} + \sin^2 \frac{4\pi}{9} = 2$

Q 5 | Page 40

Prove that:
$\sec\left( \frac{3\pi}{2} - x \right)\sec\left( x - \frac{5\pi}{2} \right) + \tan\left( \frac{5\pi}{2} + x \right)\tan\left( x - \frac{3\pi}{2} \right) = - 1 .$

Q 6.1 | Page 40

In a ∆ABC, prove that:
cos (A + B) + cos C = 0

Q 6.2 | Page 40

In a ∆ABC, prove that:

$\cos\left( \frac{A + B}{2} \right) = \sin\frac{C}{2}$

Q 6.3 | Page 40

In a ∆ABC, prove that:

$\tan\frac{A + B}{2} = \cot\frac{C}{2}$
Q 7 | Page 40

In a ∆A, B, C, D be the angles of a cyclic quadrilateral, taken in order, prove that cos(180° − A) + cos (180° + B) + cos (180° + C) − sin (90° + D) = 0

Q 8.1 | Page 40

Find x from the following equations:
$cosec\left( \frac{\pi}{2} + \theta \right) + x \cos \theta \cot\left( \frac{\pi}{2} + \theta \right) = \sin\left( \frac{\pi}{2} + \theta \right)$

Q 8.2 | Page 40

Find x from the following equations:
$x \cot\left( \frac{\pi}{2} + \theta \right) + \tan\left( \frac{\pi}{2} + \theta \right)\sin \theta + cosec\left( \frac{\pi}{2} + \theta \right) = 0$

Q 9.1 | Page 40

Prove that:
$\tan 4\pi - \cos\frac{3\pi}{2} - \sin\frac{5\pi}{6}\cos\frac{2\pi}{3} = \frac{1}{4}$

Q 9.2 | Page 40

Prove that:
$\sin\frac{13\pi}{3}\sin\frac{8\pi}{3} + \cos\frac{2\pi}{3}\sin\frac{5\pi}{6} = \frac{1}{2}$

Q 9.3 | Page 40

Prove that:
$\sin \frac{13\pi}{3}\sin\frac{2\pi}{3} + \cos\frac{4\pi}{3}\sin\frac{13\pi}{6} = \frac{1}{2}$

Q 9.4 | Page 40

Prove that:

$\sin\frac{10\pi}{3}\cos\frac{13\pi}{6} + \cos\frac{8\pi}{3}\sin\frac{5\pi}{6} = - 1$
Q 9.5 | Page 40

Prove that:

$\tan\frac{5\pi}{4}\cot\frac{9\pi}{4} + \tan\frac{17\pi}{4}\cot\frac{15\pi}{4} = 0$

Pages 40 - 41

Q 1 | Page 40

Write the maximum and minimum values of cos (cos x).

Q 2 | Page 40

Write the maximum and minimum values of sin (sin x).

Q 3 | Page 40

Write the maximum value of sin (cos x).

Q 4 | Page 40

If sin x = cos2 x, then write the value of cos2 x (1 + cos2 x).

Q 5 | Page 40

If sin x + cosec x = 2, then write the value of sinn x + cosecn x.

Q 6 | Page 40

If sin x + sin2 x = 1, then write the value of cos12 x + 3 cos10 x + 3 cos8 x + cos6 x.

Q 7 | Page 40

If sin x + sin2 x = 1, then write the value of cos8 x + 2 cos6 x + cos4 x.

Q 8 | Page 40

If sin θ1 + sin θ2 + sin θ3 = 3, then write the value of cos θ1 + cos θ2 + cos θ3.

Q 9 | Page 40

Write the value of sin 10° + sin 20° + sin 30° + ... + sin 360°.

Q 10 | Page 40

A circular wire of radius 15 cm is cut and bent so as to lie along the circumference of a loop of radius 120 cm. Write the measure of the angle subtended by it at the centre of the loop.

Q 11 | Page 41

Write the value of 2 (sin6 x + cos6 x) −3 (sin4 x + cos4 x) + 1.

Q 12 | Page 41

Write the value of cos 1° + cos 2° + cos 3° + ... + cos 180°.

Q 13 | Page 41

If cot (α + β) = 0, then write the value of sin (α + 2β).

Q 14 | Page 41

If tan A + cot A = 4, then write the value of tan4 A + cot4 A.

Q 15 | Page 41

Write the least value of cos2 x + sec2 x.

Q 16 | Page 41
If x = sin14 x + cos20  x, then write the smallest interval in which the value of x lie.
Q 17 | Page 41

If 3 sin x + 5 cos x = 5, then write the value of 5 sin x − 3 cos x.

Pages 41 - 43

Q 1 | Page 41

If tan x = $x - \frac{1}{4x}$, then sec x − tan x is equal to

$- 2x, \frac{1}{2x}$

$- \frac{1}{2x}, 2x$

2x

$2x, \frac{1}{2x}$

Q 2 | Page 41

If sec $x = x + \frac{1}{4x}$, then sec x + tan x =

$x, \frac{1}{x}$

$2x, \frac{1}{2x}$

$- 2x, \frac{1}{2x}$

$- \frac{1}{x}, x$

Q 3 | Page 41

If $\frac{\pi}{2} < x < \frac{3\pi}{2},\text{ then }\sqrt{\frac{1 - \sin x}{1 + \sin x}}$ is equal to

sec x − tan x

sec x + tan x

tan x − sec x

none of these

Q 4 | Page 41
$\sqrt{\frac{1 + \cos x}{1 - \cos x}}$ is equal to

cosec x + cot x

cosec x − cot x

−cosec x + cot x

−cosec x − cot x

Q 5 | Page 41

If $0 < x < \frac{\pi}{2}$, and if $\frac{y + 1}{1 - y} = \sqrt{\frac{1 + \sin x}{1 - \sin x}}$, then y is equal to

$\cot\frac{x}{2}$

$\tan\frac{x}{2}$

$\cot\frac{x}{2} + \tan\frac{x}{2}$

$\cot\frac{x}{2} - \tan\frac{x}{2}$

Q 6 | Page 41

If $\frac{\pi}{2} < x < \pi, \text{ then }\sqrt{\frac{1 - \sin x}{1 + \sin x}} + \sqrt{\frac{1 + \sin x}{1 - \sin x}}$ is equal to

2 sec x

−2 sec x

sec x

−sec x

Q 7 | Page 41

If x = r sin θ cos ϕ, y = r sin θ sin ϕ and r cos θ, then x2 + y2 + z2 is independent of

θ, ϕ

r, θ

r, ϕ

r

Q 8 | Page 41

If tan x + sec x = $\sqrt{3}$, 0 < x < π, then x is equal to

$\frac{5\pi}{6}$

$\frac{2\pi}{3}$

$\frac{\pi}{6}$

$\frac{\pi}{3}$
Q 9 | Page 41

If tan $x = - \frac{1}{\sqrt{5}}$ and θ lies in the IV quadrant, then the value of cos x is

$\frac{\sqrt{5}}{\sqrt{6}}$

$\frac{2}{\sqrt{6}}$

$\frac{1}{2}$

$\frac{1}{\sqrt{6}}$

Q 10 | Page 42

If $\frac{3\pi}{4} < \alpha < \pi, \text{ then }\sqrt{2\cot \alpha + \frac{1}{\sin^2 \alpha}}$ is equal to

1 − cot α

1 + cot α

−1 + cot α

−1 −cot α

Q 11 | Page 42

sin6 A + cos6 A + 3 sin2 A cos2 A =

0

1

2

3

Q 12 | Page 42

If $cosec x - \cot x = \frac{1}{2}, 0 < x < \frac{\pi}{2},$

$\frac{5}{3}$

$\frac{3}{5}$

$- \frac{3}{5}$

$- \frac{5}{3}$

Q 13 | Page 42

If $cosec x + \cot x = \frac{11}{2}$, then tan x =

$\frac{21}{22}$

$\frac{15}{16}$

$\frac{44}{117}$

$\frac{117}{44}$

Q 14 | Page 42
$\sec^2 x = \frac{4xy}{(x + y )^2}$ is true if and only if

x + y ≠ 0

x = y, x ≠ 0

x = y

x ≠0, y ≠ 0

Q 15 | Page 42

If x is an acute angle and $\tan x = \frac{1}{\sqrt{7}}$, then the value of $\frac{{cosec}^2 x - \sec^2 x}{{cosec}^2 x + \sec^2 x}$ is

3/4

1/2

2

5/4

Q 16 | Page 42

The value of sin25° + sin210° + sin215° + ... + sin285° + sin290° is

7

8

9.5

10

Q 17 | Page 42

sin2 π/18 + sin2 π/9 + sin2 7π/18 + sin2 4π/9 =

1

4

2

0

Q 18 | Page 42

If tan A + cot A = 4, then tan4 A + cot4 A is equal to

110

191

80

194

Q 19 | Page 42

If x sin 45° cos2 60° = $\frac{\tan^2 60^\circ cosec30^\circ}{\sec45^\circ \cot^{2^\circ} 30^\circ}$, then x =

2

4

8

16

Q 20 | Page 42

If A lies in second quadrant 3tan A + 4 = 0, then the value of 2cot A − 5cosA + sin A is equal to

$- \frac{53}{10}$

$\frac{23}{10}$

$\frac{37}{10}$

$\frac{7}{10}$

Q 21 | Page 42

If $cosec x + \cot x = \frac{11}{2}$, then tan x =

$\frac{21}{22}$

$\frac{15}{16}$

$\frac{44}{117}$

$\frac{117}{43}$

Q 22 | Page 42

If tan θ + sec θ =ex, then cos θ equals

$\frac{e^x + e^{- x}}{2}$

$\frac{2}{e^x + e^{- x}}$

$\frac{e^x - e^{- x}}{2}$

$\frac{e^x - e^{- x}}{e^x + e^{- x}}$

Q 23 | Page 42

If sec x + tan x = k, cos x =

$\frac{k^2 + 1}{2k}$

$\frac{2k}{k^2 + 1}$

$\frac{k}{k^2 + 1}$

$\frac{k}{k^2 - 1}$

Q 24 | Page 43

If $f\left( x \right) = \cos^2 x + \sec^2 x$, then

f(x) < 1

f(x) = 1

1 < f(x) < 2

f(x) ≥ 2

Q 25 | Page 43

Which of the following is incorrect?

$\sin x = - \frac{1}{5}$

cos x = 1

$\sec x = \frac{1}{2}$

tan x = 20

Q 26 | Page 43

The value of $\cos1^\circ \cos2^\circ \cos3^\circ . . . \cos179^\circ$ is

$\frac{1}{\sqrt{2}}$

0

1

-1

Q 27 | Page 43

The value of $\tan1^\circ \tan2^\circ \tan3^\circ . . . \tan89^\circ$ is

0

1

$\frac{1}{2}$

not defined

Q 28 | Page 43

Which of the following is correct?

$\sin1^\circ > \sin1$

$\sin1^\circ < \sin1$

$\sin1^\circ = \sin1$

$\sin1^\circ = \frac{\pi}{180}\sin1$

RD Sharma solutions for Class 11 Mathematics chapter 5 - Trigonometric Functions

RD Sharma solutions for Class 11 Maths chapter 5 (Trigonometric Functions) 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 5 Trigonometric Functions 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 Trigonometric Functions 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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