Question

How do you verify the identity `(cot^2x-1)/(1+cot^2x)=1-2sin^2x`?

Answer 1

see explanation.

Explanation:

We attempt to express the left side in the same form as the right side.
Let's begin by rewriting `cot^2x`

`color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(cotx=(cosx)/(sinx))color(white)(a/a)|)))`

`rArr((cos^2x)/(sin^2x)-1)/(1+cos^2x/(sin^2x)`

now multiply all terms on numerator and denominator by `sin^2x`

`rArr(cos^2x-sin^2x)/(sin^2x+cos^2x) .......(A)`

To simplify we require the `color(blue)"trigonometric identities"`

`color(orange)"Reminder"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(cos2x=cos^2x-sin^2x=1-2sin^2x)color(white)(a/a)|)))`

and `color(red)(|bar(ul(color(white)(a/a)color(black)(sin^2x+cos^2x=1)color(white)(a/a)|)))`

Substituting these into (A) gives.

`cos2x=1-2sin^2x`

Thus left side = right side `rArr" verified"`