Trigonometry/Identities

Let us take a right angled triangle with hypotenuse length 1. If we mark one of the acute angles as $$\theta$$, then using the definition of the sine ratio, we have
 * $$\sin \theta = \cfrac{opposite}{hypotenuse}$$

As the hypotenuse is 1,
 * $$\sin \theta = \cfrac{opposite}{1} = opposite$$

Repeating the same process using the definition of the cosine ratio, we have
 * $$\cos \theta = \cfrac{adjacent}{hypotenuse} = \cfrac{adjacent}{1} = adjacent$$

Pythagorean identities
Since this is a right triangle, we can use the Pythagorean Theorem:
 * $$ x^2 + y^2 = r^2$$
 * $$ \frac{x^2}{r^2} + \frac{y^2}{r^2} = \frac{r^2}{r^2} $$
 * $$\operatorname{cos}^2 \theta + \operatorname{sin}^2 \theta = 1$$

This is the most fundamental identity in trigonometry.
 * $$ \frac{x^2}{y^2} + \frac{y^2}{y^2} = \frac{r^2}{y^2} $$
 * $$\operatorname{cot}^2 x + 1 = \operatorname{csc}^2$$


 * $$\frac{x^2}{x^2} + \frac{y^2}{x^2} = \frac{r^2}{x^2}$$
 * $$\operatorname 1 + \operatorname{tan}^2\theta = \operatorname{sec}^2\theta$$

From this identity, if we divide through by squared cosine, we are left with:
 * $$\cfrac{\operatorname{sin}^2 \theta + \operatorname{cos}^2 \theta}{\operatorname{cos}^2 \theta} = \cfrac{1}{\operatorname{cos}^2 \theta}$$
 * $$\operatorname{tan}^2 \theta + 1 = \operatorname{sec}^2\theta$$
 * $$\operatorname{sec}^2 \theta - \operatorname{tan}^2 \theta = 1$$

If instead we divide the original identity by squared sine, we are left with:
 * $$\cfrac{\operatorname{sin}^2 \theta + \operatorname{cos}^2 \theta}{\operatorname{sin}^2 \theta} = \cfrac{1}{\operatorname{sin}^2 \theta}$$
 * $$\operatorname{cot}^2 \theta + 1 = \operatorname{csc}^2 \theta$$
 * $$\operatorname{csc}^2 \theta - \operatorname{cot}^2 \theta = 1$$

There are basically 3 main trigonometric identities. The proofs come directly from the definitions of these functions and the application of the Pythagorean theorem:
 * $$\operatorname{sin}^2 \theta + \operatorname{cos}^2 \theta = 1$$


 * $$\operatorname{sec}^2 \theta - \operatorname{tan}^2 \theta = 1$$


 * $$\operatorname{csc}^2 \theta - \operatorname{cot}^2 \theta = 1$$

Angle sum-difference identities

 * $$\sin(\alpha \pm \beta)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta$$
 * $$\cos(\alpha \pm \beta)=\cos \alpha\cos \beta\mp \sin \alpha \sin \beta$$

Cofunction identities

 * $$\cos(90-\theta)=\sin\theta$$
 * $$\sec(90-\theta)=\csc\theta$$
 * $$\tan(90-\theta)=\cot\theta$$


 * $$\sin(90-\theta)=\cos\theta$$
 * $$\csc(90-\theta)=\sec\theta$$
 * $$\cot(90-\theta)=\tan\theta$$

Multiple angle identities

 * $$\cos2A=\cos^2A-\sin^2A$$
 * $$\sin2A=2\sin A\cos A$$
 * $$\sin 2\theta= \frac{tan 2\theta*tan\theta}{tan 2\theta-tan\theta} $$
 * $$\cos 2\theta=\frac{tan \theta}{tan 2\theta - tan\theta}$$
 * $$\tan 2\theta=tan\theta(\frac{1}{cos 2\theta}+1)$$
 * $$\tan 2\theta=\frac{2sin^2\theta}{sin 2\theta-tan\theta}$$