List of Trigonometric Identities

• Given below is the list of 24 trigonometric identities.
• Derivation of the first nine identities can be seen here.
• Derivation of the remaining identities can be seen here.

Identity 1: sin (-x) = -sin x  

Identity 2: cos (-x) = cos x 

Identity 3: cos (x+y) = cos x cos y - sin x sin y 

Identity 4: cos (x-y) = cos x cos y + sin x sin y 

Identity 5: $\cos (\frac{\pi}{2} - x)= \sin x$

Identity 6: $ \sin \left(\frac{\pi}{2} - x\right)=\cos x$

Identity 7: $\sin (x+y)=\sin x \, \cos y + \cos x \sin y$

Identity 8: $\sin (x-y)=\sin x \, \cos y - \cos x \sin y $

Identity 9(a): $\cos (\frac{\pi}{2}+x) =- \sin x$

Identity 9(b): $\sin (\frac{\pi}{2}+x) = \cos x$ 

Identity 9(c): cos (π - x) = - cos x 

Identity 9(d): sin (π - x) = sin x 

Identity 9(e): cos (π + x) = -cos x  

Identity 9(f): sin (π + x) = -sin x 

Identity 9(g): cos (2π - x) = cos x.

Identity 9(h): sin (2π - x) = -sin x 

Identity 10: $\tan (x+y)=\frac{\tan x + \tan y}{1 - \tan x \tan y}$

Identity 11: $\tan (x-y)=\frac{\tan x - \tan y}{1 + \tan x \tan y}$

Identity 12: $\cot (x+y)=\frac{\cot x \cot y - 1}{\cot y + \cot x}$

Identity 13: $\cot (x-y)=\frac{\cot x \cot y + 1}{\cot y - \cot x}$

Identity 14:
$\cos 2x \;\;= \cos^2x-\sin^2x\;\;=1-2\sin^2x\;\;=2cos^2x-1\;\;=\frac{1-\tan^2x}{1+\tan^2x}$

Identity 15: $\sin 2x \;\;= 2\sin x \cos x \;\;= \frac{2\tan x}{1+\tan^2x}$

Identity 16: $\tan 2x=\frac{2\tan x}{1 - \tan^2 x}$ 

Identity 17: sin 3x = 3 sin x  - 4 sin³x

Identity 18: cos 3x = 4 cos³x - 3 cos x

Identity 19: $\tan 3x=\frac{3\tan x - \tan^3x}{1 - 3\tan^2}$

Identity 20(a): $\cos x + \cos y = 2 \cos \frac{x + y}{2} \cos \frac{x - y}{2}$

Identity 20(b): $\cos x - \cos y = -2 \sin \frac{x + y}{2} \sin \frac{x - y}{2}$

Identity 20(c): $\sin x + \sin y = 2 \sin \frac{x + y}{2} \cos \frac{x - y}{2}$ 

Identity 20(d): $\sin x - \sin y = 2 \cos \frac{x + y}{2} \sin \frac{x - y}{2}$

Identity 21(a): $2 \cos x \cos y=\cos (x+y) + \cos (x-y)$

Identity 21(b): $-2 \sin x \sin y=\cos (x+y) - \cos (x-y)$

Identity 21(c): $2 \sin x \cos y=\sin (x+y) + \sin (x-y)$

Identity 21(d): $2 \cos x \sin y=\sin (x+y) - \sin (x-y)$

Identity 22(a): $\sin x ~=~ \pm \sqrt{\frac{1 - \cos 2x}{2}}$

Identity 22(b): $\sin \frac{x}{2} ~=~ \pm \sqrt{\frac{1 - \cos x}{2}}$

Identity 23(a): $\cos x ~=~ \pm \sqrt{\frac{1 + \cos 2x}{2}}$

Identity 23(b): $\cos \frac{x}{2} ~=~ \pm \sqrt{\frac{1 + \cos x}{2}}$

Identity 24(a): $\tan x ~=~\frac{1 - \cos 2x}{\sin 2x}$

Identity 24(b): $\tan x ~=~ \frac{\sin 2x}{1 + \cos 2x}$

Identity 24(c): $\tan \frac{x}{2} ~=~ \frac{1 - \cos x}{\sin x}$

Identity 24(d): $\tan \frac{x}{2} ~=~ \frac{\sin x}{1 + \cos x}$

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Solved examples related to the above identities can be seen here.

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