Intro to Trig Ratios and to Identity
Simplifying Trig Expression ex1
Simplifying Trig Expression ex2
Simplifying Trig Expression ex3
Simplifying Trig Expression ex4
Introduction to Trig Identities
ex1 Proving identity
ex2 Proving identity
ex3 Proving identity
Compounded Angle ex1 identity
Compounded Angle ex2 identity
Compounded Angle ex3 identity
Prove \displaystyle \cos^4 x - \sin^4 x = 1- 2\sin^2x
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Prove \displaystyle \csc^2 x + \sec^2 x = \csc^2\sec^2x
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Prove \displaystyle \cos^2 x \cos^2 y + \sin^2x\sin^2y + \sin^2x\cos^2y + \sin^2y\cos^2x = 1
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Prove \displaystyle \sec^2x - \sec^2y = \tan^2x -\tan^2y
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Prove \displaystyle \frac{\tan x+ \tan y}{\cot x + \cot y} = (\tan x)(\tan y)
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Prove \displaystyle (\sec x -\cos x)(\csc x - \sin x) =\frac{\tan x}{1+\tan^2x}
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Prove \displaystyle \cos^6x + \sin^6x = 1 - 3\sin^2x+ 3\sin^4x
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Prove \displaystyle \sec^6x - \tan^6x = 1 + 3\tan^2x \sec^2x
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Prove \displaystyle 1 + \cot x \tan y = \frac{\sin(x+ y)}{\sin x \cos y}
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Prove \displaystyle \cos(x+y)\cos y + \sin(x+y)\sin y = \cos x
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Prove \displaystyle \sin x- \tan y \cos x = \frac{\sin(x-y)}{\cos y}
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Prove \displaystyle \sin(x +y)\sin(x -y) = \cos^2y - \cos^2x
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Prove \displaystyle \tan(x+y)\tan(x-y) = \frac{\sin^2x - \sin^2y}{\cos^2x-\sin^2y}
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Prove \displaystyle
\frac{\tan(x-y)+ \tan y}{1-\tan(x-y)\tan y} = \tan x
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Prove \displaystyle \sin 5x = \sin x(\cos^22x-\sin^22x) + 2\cos x \cos 2x \sin 2x
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Prove \displaystyle \sin(\frac{\pi}{2} - x)\cot(\frac{\pi}{2}+x) = - \sin x
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Prove \displaystyle \cos(-x) + \cos(\pi- x) = \cos(\pi + x)+ \cos x
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Prove \displaystyle \frac{\sin 2x}{1+ \cos 2x} = \tan x
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Prove \displaystyle \frac{1 + \cos x}{\sin x} = \cot \frac{x}{2}
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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}