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
$$(cosecx-\sin x)(\sec x-\cos x)(\tan x+\cot x)=1$$

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

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

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
$${(\sec x\sec y+\tan x\tan y)}^2-{(\sec x\tan y+\tan x\sec y)}^2=1$$

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
$${(1+\tan \alpha \tan \beta )}^2+{(\tan \alpha -\tan \beta )}^2={\sec }^2\alpha {\sec }^2\beta$$

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 $$cosecx-\sin x=a^3,\sec x-\cos x=b^3$$, then prove that $$a^2{b}^2(a^2+b^2)=1$$

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

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

If $$a=\sec x-\tan x\text{ and }b=cosecx+\cot x$$, 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  $$2T_6-3T_4+1=0$$

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

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 $$\frac{cosecx+\cot x}{\sec x-\tan x}$$

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

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:
$$cosec(\frac{\pi }{2}+\theta )+x\cos \theta \cot (\frac{\pi }{2}+\theta )=\sin (\frac{\pi }{2}+\theta )$$

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

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

If $$cosecx+\cot x=\frac{11}{2}$$, 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° = $$\frac{{\tan }^2{60}^{\circ }cosec{30}^{\circ }}{\sec {45}^{\circ }{\cot }^{2^{\circ }}{30}^{\circ }}$$, 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 $$cosecx+\cot x=\frac{11}{2}$$, 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 $$\cos 1^{\circ }\cos 2^{\circ }\cos 3^{\circ }...\cos {179}^{\circ }$$ is

The value of $$\tan 1^{\circ }\tan 2^{\circ }\tan 3^{\circ }...\tan {89}^{\circ }$$ is

Which of the following is correct?