Question

(1−cos^2 x )·(1+tan^2 x)

Answer 1

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

Explanation:

Given

`(1-cos^2 x ).(1+tan^2 x)`

Required

Solve

`(1-cos^2 x ).(1+tan^2 x)`

In trigonometry;

`1 - cos^2x = sin^2x`

So, make substitution

`(1-cos^2 x ).(1+tan^2 x)` becomes

`(1-cos^2 x ).(1+tan^2 x) = (sin^2 x ).(1+tan^2 x)`

Also; in trigonometry:

`1 + tan^2x = sec^2x`

Make another substitution

`(1-cos^2 x ).(1+tan^2 x) = (sin^2 x ).(sec^2 x)`

Recall that
`secx = (1)/(cosx)`

So;

`(1-cos^2 x ).(1+tan^2 x) = (sin^2 x ).(sec^2 x)` becomes

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

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

`(1-cos^2 x ).(1+tan^2 x) = ((sin x )/(cosx))^2`

In trigonometry;

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

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

The expression cannot be further simplified