Question

Verify each identity. 1 - cos 2x/ tan^2 x = 1 + cos 2 x

Answer 1

Identity:

`(1-\cos2x)/(\tan^2x)=1+\cos 2x`

We can also write the given identity as :

`(1-\cos 2x)/(1+\cos 2x)=\tan ^2x`

From the trignometric ratio of tanx we have:

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

and

`\begin{gathered} \text{Cos}2x=1-2\sin ^2x \\ \Rightarrow1-\cos 2x=2\sin ^2x \\ \text{Cos}2x=2\cos ^2x-1 \\ \Rightarrow1+\cos 2x=2\cos ^2x \end{gathered}`

So, substitute the value and simplify:

`\begin{gathered} (1-\cos2x)/(1+\cos2x)=\tan ^2x \\ (2\sin^2x)/(2\cos^2x)=\tan ^2x \\ (\sin ^2x)/(\cos ^2x)=\tan ^2x \\ \text{tan}^2x=\tan ^2x \end{gathered}`

So, LHS = RHS, the identity is proved