Question

How do you factor and simplify `tan^2x-cot^2x`?

Answer 1

`-(4(sin x - cos x)(sin x + cos x))/(sin^2 2x)` or
-4cot 2x.csc 2x

Explanation:

`tan^2 x - cot^2 x = sin^2 x/(cos^2 x) - cos^2 x/(sin^2 x) =`
`= (sin^4 x - cos^4 x)/(sin^2 x.cos ^2 x)=`
Since:
`(a) = (sin^4 x - cos^4 x) = (sin ^2 x - cos^2 x)(sin^2 x + cos^2 x) =`
`= (sin x - cos x)(sin x + cos x)` , and
`(b) = sin^2 x.cos^2 x = (1/4)sin^2 2x`,
There for:
`tan^2 x - cot^2 x = ((a))/((b)) = (4(sin x - cos x)(sin x + cos x))/(sin^2 2x)`

There is another answer for simplification:
Since (sin^2 x - cos^2 x) = - cos 2x, then,
`((a))/((b)) = - (4cos 2x)/(sin^2 2x) = - (4 cot 2x)(1/(sin 2x)) =`
`- 4cot 2x.csc 2x` #