Double Angle Formulas

Theorem

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

where $$\sin, \cos, \tan$$ are sine, cosine and tangent.

Corollary

 * $$\cos(2\theta) = 2 \cos^2\theta - 1$$
 * $$\cos(2\theta) = 1 - 2 \sin^2\theta$$

Proof
$$ $$ $$ $$ $$

We then equate real and imaginary parts:

$$ $$

Since $$\tan\theta=\frac{\sin\theta}{\cos\theta}$$, we have:

$$\tan(2\theta)=\frac{2\sin\theta\cos\theta}{\cos^2\theta-\sin^2\theta}$$ which is equal to $$\frac{2\tan\theta}{1-\tan^2\theta}$$ by dividing the top and bottom by $$\cos^2\theta$$

Proof of Corollary
$$ $$ $$

$$ $$ $$

Alternative Proof
$$ $$ $$

$$ $$ $$

$$ $$ $$