Trigonometric Identities

Collecting useful trigonometric substitution and parametrization identities.

Standard Identities

1. $si{n}^2+co{s}^2=1$
2. $si{n}^2(x)=\frac{1-cos(2x)}{2}$
3. $co{s}^2(x)=\frac{1+cos(2x)}{2}$
4. $si{n}^2(x)co{s}^2(x)=\frac{1}{4}si{n}^2(2x)=\frac{1}{8}(1-cos(4x))$
5. $cos(\alpha )cos(\beta )=\frac{1}{2}\text{[}cos(\alpha -\beta )+cos(\alpha +\beta )]$
6. $sin(\alpha )cos(\beta )=\frac{1}{2}\text{[}sin(\alpha -\beta )+(sin(\alpha +\beta )]$
7. $sin(\alpha )sin(\beta )=\frac{1}{2}\text{[}cos(\alpha -\beta )-cos(\alpha +\beta )]$

Differential Identities

1. $\frac{d}{dx}(1+si{n}^{2}x)=2sinxcosx$
2. $\frac{d}{dx}(si{n}^{2}x-co{s}^{2}x)=4sinxcosx$