Trigonometry : Trigonometry identities

Pythagorian theorem :

$$si{n}^2(\theta )+co{s}^2(\theta )=1$$

$$(1)\times \frac{1}{si{n}^2(\theta )},1+co{t}^2(\theta )=cose{c}^2(\theta )$$
$$(1)\times \frac{1}{co{s}^2(\theta )},ta{n}^2(\theta )+1=se{c}^2(\theta )$$

Compound Angle :

$$cos(A+B)=cos(A)cos(B)-sin(A)sin(B)$$
$$cos(A-B)=cos(A)cos(B)+sin(A)sin(B)$$
$$sin(A+B)=sin(A)cos(B)+sin(B)cos(A)$$
$$sin(A-B)=sin(A)cos(B)-sin(B)cos(A)$$
$$tan(A+B)=\frac{tan(A)+tan(B)}{1-tan(A)tan(B)}$$
$$tan(A-B)=\frac{tan(A)-tan(B)}{1+tan(A)tan(B)}$$
$$cos(2\theta )=co{s}^2(\theta )-si{n}^2(\theta )=2co{s}^2(\theta )-1=1-2si{n}^2(\theta )$$
$$sin(2\theta )=2sin(\theta )cos(\theta )$$
$$tan(2\theta )=\frac{2tan(\theta )}{1-ta{n}^2(\theta )}$$

Half angle tangent substitution :

$$sin(\theta )=\frac{2tan(\frac{\theta }{2})}{1+tan(\frac{\theta }{2})}$$
$$cos(\theta )=\frac{1-ta{n}^2(\frac{\theta }{2})}{1+ta{n}^2(\frac{\theta }{2})}$$
$$tan(\theta )=\frac{2tan(\frac{\theta }{2})}{1+ta{n}^2(\frac{\theta }{2})}$$