Prove \displaystyle \cos^4 x - \sin^4 x = 1- 2\sin^2x.

Prove \displaystyle \csc^2 x + \sec^2 x = \csc^2\sec^2x.

Prove \displaystyle \cos^2 x \cos^2 y + \sin^2x\sin^2y + \sin^2x\cos^2y + \sin^2y\cos^2x = 1.

Prove \displaystyle \sec^2x - \sec^2y = \tan^2x -\tan^2y.

Prove \displaystyle \frac{\tan x+ \tan y}{\cot x + \cot y} = (\tan x)(\tan y).

Prove \displaystyle (\sec x -\cos x)(\csc x - \sin x) =\frac{\tan x}{1+\tan^2x}.

Prove \displaystyle \cos^6x + \sin^6x = 1 - 3\sin^2x+ 3\sin^4x.

Prove \displaystyle \sec^6x - \tan^6x = 1 + 3\tan^2x \sec^2x.

Prove \displaystyle 1 + \cot x \tan y = \frac{\sin(x+ y)}{\sin x \cos y}.

Prove \displaystyle \cos(x+y)\cos y + \sin(x+y)\sin y = \cos x.

Prove \displaystyle \sin x- \tan y \cos x = \frac{\sin(x-y)}{\cos y}.

Prove \displaystyle \sin(x +y)\sin(x -y) = \cos^2y - \cos^2x.

Prove \displaystyle \tan(x+y)\tan(x-y) = \frac{\sin^2x - \sin^2y}{\cos^2x-\sin^2y} .

Prove \displaystyle \frac{\tan(x-y)+ \tan y}{1-\tan(x-y)\tan y} = \tan x .

Prove \displaystyle \sin 5x = \sin x(\cos^22x-\sin^22x) + 2\cos x \cos 2x \sin 2x.

Prove \displaystyle \sin(\frac{\pi}{2} - x)\cot(\frac{\pi}{2}+x) = - \sin x.

Prove \displaystyle \cos(-x) + \cos(\pi- x) = \cos(\pi + x)+ \cos x.

Prove \displaystyle \frac{\sin 2x}{1+ \cos 2x} = \tan x.

Prove \displaystyle \frac{1 + \cos x}{\sin x} = \cot \frac{x}{2}.

Prove \displaystyle 2\csc 2x = \sec x \csc x

Prove \displaystyle 2 \cot 2x = \cot x - \tan x

Prove \displaystyle \frac{\cos 2x}{1+ \sin 2x} = \tan(\frac{\pi}{4} - x)

Prove \displaystyle \frac{\cos x - \sin x}{\cos x + \sin x} = \sec 2x - \tan 2x

Prove \displaystyle \frac{1 - \cos 2x + \sin 2x}{1 + \cos 2x + \sin 2x} = \tan x

Prove \displaystyle \cos^6x -\sin^6x = \cos 2x(1 - \frac{1}{4}\sin^22x)

Prove \displaystyle 4(\cos^6x + \sin^6x) = 1 + 3\cos^2 2x

Prove \displaystyle \sec x - \tan x = \tan(\frac{\pi}{4}- \frac{x}{2})

Prove \displaystyle \frac{\sin 2x}{1+\cos 2x} \times \frac{\cos x}{1 + \cos x} = \tan \frac{x}{2}

Prove \displaystyle \sin^2x + \cos^4x = \cos^2 x + \sin^4 x

Prove \displaystyle \tan x - \cot x = (\tan x -1)(\cot x + 1)

Prove \displaystyle \cos x = \sin x \tan^2x\cot^3x

Prove \displaystyle (\sin x + \cos x)(\tan x + \cot x) = \sec x + \csc x

Prove \displaystyle \sin^4x + \cos^4x = \sin^2x(\csc^2x - 2\cos^2 x)

Prove \displaystyle \sin^3x + \cos^3x = (1- \sin x \cos x)(\sin x + \cos x)

Prove \displaystyle \cos(\frac{\pi}{12} -x)\sec\frac{\pi}{12} - \sin(\frac{\pi}{12} -x)\csc\frac{\pi}{12} = 4\sin x

Prove \displaystyle \tan(x-y)+ \tan(y - z) = \frac{\sec^2y(\tan x - \tan z)}{(1 + \tan x \tan z)(1+\tan y\tan z)}

Prove \displaystyle \sin 8x = 8\sin x \cos x \cos 2x \cos 4x

Prove \displaystyle \sin x = 1 - 2\sin^2(\frac{\pi}{4} - \frac{x}{2})

Prove \displaystyle \sin( x + y) + \sin (x - y) = 2\sin x \cos y

Prove \displaystyle \frac{\sin(x-y)}{\sin x \sin y} + \frac{\sin(y -z)}{\sin y \sin z} + \frac{\sin(z -x)}{\sin z \sin x}= 0

Prove \displaystyle \tan x+ \tan(\pi -x) + \cot(\frac{\pi}{2} + x) = \tan(2\pi -x)

Prove \displaystyle \tan(x + y + z) = \frac{\tan x \tan y \tan z - \tan x \tan y \tan z }{1 - \tan x \tan y - \tan x \tan z - \tan y \tan z}

Prove \displaystyle \csc^2(\frac{\pi}{2} - x) = 1 + \sin^2x\csc^2(\frac{\pi}{2} - x)

Prove \displaystyle \tan(\frac{\pi}{4} + x) + \tan(\frac{\pi}{4} - x) = 2\sec 2x

Prove \displaystyle \frac{1 -\sin 2x}{\cos 2x} = \frac{\cos 2x}{1+ \sin 2x}