Trigonometric Identities

Half Angle Identities

$$\cos \frac{\theta }{2}=\sqrt{\frac{1+\cos \theta }{2}}$$

$$\sin \frac{\theta }{2}=\sqrt{\frac{1-\cos \theta }{2}}$$

therefore:

$$2{\cos }^2\frac{\theta }{2}=1+\cos \theta$$

$$2{\sin }^2\frac{\theta }{2}=1-\cos \theta$$

Sum and Difference Formulas

$$\cos (a+b)=\cos a\cos b-\sin a\sin b$$

$$\cos (a-b)=\cos a\cos b+\sin a\sin b$$

$$\sin (a+b)=\sin a\cos b+\cos a\sin b$$

$$\sin (a-b)=\sin a\cos b-\cos a\sin b$$

Linear Combinations

$$a\cos x+b\sin x=\sqrt{a^2+b^2}\cos (x-\arctan 2(b,a))$$

$$a\cos x+b\sin x=\sqrt{a^2+b^2}\cos (x+\mathrm{Arctan}(-b/a)))$$