01 Trignometric
| Rule | |||
|---|---|---|---|
| Product | \(2 \sin x \cos y\) | \(\sin(x+y) + \sin(x-y)\) | |
| \(2 \cos x \sin y\) | \(\sin(x+y) - \sin(x-y)\) | ||
| \(2 \cos x \cos y\) | \(\cos(x+y) + \cos(x-y)\) | ||
| \(2 \sin x \sin y\) | \(\cos(x-y) - \cos(x+y)\) | ||
| Sum | \(\sin C + \sin D\) | \(2 \sin \left( \frac{C+D}{2} \right) \cos \left( \frac{C-D}{2} \right)\) | |
| \(\sin C - \sin D\) | \(2 \cos \left( \frac{C+D}{2} \right) \sin \left( \frac{C-D}{2} \right)\) | ||
| \(\cos C + \cos D\) | \(2 \cos \left( \frac{C+D}{2} \right) \cos \left( \frac{C-D}{2} \right)\) | ||
| \(\cos C - \cos D\) | \(-2 \sin \left( \frac{C+D}{2} \right) \sin \left( \frac{C-D}{2} \right)\) | ||
| \(\tan(x+y)\) | \(\frac{\tan x + \tan y}{1 - \tan x \tan y}\) | ||
| \(\tan(x-y)\) | \(\frac{\tan x - \tan y}{1 + \tan x \tan y}\) | ||
| Double | \(\sin 2x\) | \(2 \sin x \cos x\) | \(\frac{2 \tan x}{1 + \tan^2 x}\) |
| \(\cos 2x\) | \(\cos^2 x - \sin^2 x\) \(2\cos^2 x - 1\) \(1 - 2 \sin^2 x\) | \(\frac{1-\tan^2 x}{1 + \tan^2 x}\) | |
| \(\tan 2x\) | \(\frac{2 \tan x}{1 - \tan^2 x}\) | ||
| \(\cot 2x\) | \(\frac{\cot^2 x - 1}{2 \cot x}\) | ||
| Triple | \(\sin 3x\) | \(3 \sin x - 4\sin^3 x\) | |
| \(\cos 3x\) | \(4\cos^3 x - 3\cos x\) | ||
| \(\tan 3x\) | \(\frac{3\tan x - \tan^3 x}{1 - 3 \tan^2 x}\) | ||
| \(\cot 3x\) | \(\frac{3 \cot x - \cot^3 x}{1 - 3 \cot^2 x}\) | ||
| Half | \(\tan \frac{x}{2}\) | \(\text{cosec }x - \cot x\) | \(\sqrt{\frac{1-\cos x}{1+\cos x}}\) |
| \(\cot \frac{x}{2}\) | \(\text{cosec }x + \cot x\) | \(\sqrt{\frac{1+\cos x}{1-\cos x}}\) |