# How do you verify the identity tan2theta=2/(cottheta-tantheta)?

Dec 22, 2016

Rewrite $\tan \theta$ and $\cot \theta$ as sines and cosines using color(magenta)(tan theta = sintheta/costheta and cot theta = costheta/sintheta.

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

I would recommend you simplify the right hand side prior to expanding the left.

(sin2theta)/(cos2theta) = 2/((cos^2theta - sin^2theta)/(costhetasintheta)

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

We know this is true because $\sin 2 \theta = 2 \sin \theta \cos \theta$ and $\cos 2 \theta$ can be written as ${\cos}^{2} \theta - {\sin}^{2} \theta$.

Practice exercises:

1. Prove the following trig identities:

a) $\frac{{\sin}^{2} \theta + {\cos}^{2} \theta + {\cot}^{2} \theta}{1 + {\tan}^{2} \theta} = {\cot}^{2} \theta$

b) $\cos \left(x + y\right) + \cos \left(x - y\right) = 2 \cos x \cos y$

c) $\csc \left(2 \alpha\right) - \cot \left(2 \alpha\right) = \tan \alpha$

Solve the following equation for x in the interval 0 ≤ x ≤ 2pi:

$\cos \left(2 x\right) = 2 {\sin}^{2} x$

Hopefully this helps, and good luck!