RD Sharma solutions for Mathematics [English] Class 11 chapter 5 - Trigonometric Functions [Latest edition]

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

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

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

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:

Prove the following identities 

Prove the following identities

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

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

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 \[6 T_{10} - 15 T_8 + 10 T_6 - 1 = 0\]

Find the value of the other five trigonometric functions 

Find the value of the other five trigonometric functions 

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

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

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

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 \[x - \sqrt{5} \sec y\]

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 \[\frac{cosec x + \cot x}{\sec x - \tan x}\]

Find the value of the following trigonometric ratio:

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

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

Find the value of the following trigonometric ratio:

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

Find the values of the following trigonometric ratio:

Find the values of the following trigonometric ratio:

Find the values of the following trigonometric ratio:

Find the values of the following trigonometric ratio:

Find the values of the following trigonometric ratio:

Find the values of the following trigonometric ratio:

Find the values of the following trigonometric ratio:

Find the values of the following trigonometric ratio:

Find the values of the following trigonometric ratio:

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

Prove that:

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

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

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

Prove that:

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

Prove that

Prove that

Prove that

Prove that

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

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

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

In a ∆ABC, prove that:

In a ∆ABC, prove that:

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

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

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

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

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

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

Prove that:

Prove that:

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

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

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

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

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

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

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

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

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

sin⁶ A + cos⁶ A + 3 sin² A cos² A =

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

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

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

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

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

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

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

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

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

If tan θ + sec θ =e^x, then cos θ equals

If sec x + tan x = k, cos x =

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

Which of the following is incorrect?

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

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

Which of the following is correct?