Question

How can I solve this trig identity manipulating tan instead of the right side?

Answer 1

Answer:

Trig identities

$\tan(\theta)=\dfrac{\sin(\theta)}{\cos(\theta)}$

$\sin(\thet2\theta)=2 \sin(\theta)\cos(\theta)$

$\begin{aligned}\cos(\thet2\theta) & =1-2\sin^2(\theta)\\2\sin^2(\theta) & = 1-\cos(2\theta) \end{aligned}$

Solution

$\tan(\theta)=\dfrac{\sin(\theta)}{\cos(\theta)}$

$\textsf{Multiply by}\:\dfrac{2\sin(\theta)}{2\sin(\theta)}:$

$\begin{aligned}\implies \tan(\theta) & =\dfrac{\sin(\theta)}{\cos(\theta)} \times \dfrac{2\sin(\theta)}{2\sin(\theta)}\\\\ & =\dfrac{2\sin^2(\theta)}{2\sin(\theta)\cos(\theta)}\\\\ & = \dfrac{1-\cos(2\theta)}{\sin(2\theta)}\end{aligned}$

Therefore,

$\implies \dfrac{1-\cos(2\theta)}{\sin(2\theta)}=\dfrac{1-\cos(2\theta)}{\sin(2\theta)}$