Question

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

Answer 1

Trig identities

`\tan(\theta)=(\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)=(\sin(\theta))/(\cos(\theta))`

`\textsf{Multiply by}\:(2\sin(\theta))/(2\sin(\theta)):`

`\begin{aligned}\implies \tan(\theta) & =(\sin(\theta))/(\cos(\theta)) * (2\sin(\theta))/(2\sin(\theta))\\\\ & =(2\sin^2(\theta))/(2\sin(\theta)\cos(\theta))\\\\ & = (1-\cos(2\theta))/(\sin(2\theta))\end{aligned}`

Therefore,

`\implies (1-\cos(2\theta))/(\sin(2\theta))=(1-\cos(2\theta))/(\sin(2\theta))`