Question

How do you prove `sec^4(x)-csc^2(x)=tan^2(x)-cot^2(x)`?

Answer 1

See explanation.

Explanation:

Are you sure you didn't mean `sec^2x-csc^2x=tan^2x-cot^2x`? I don't think the data you provided is an identity. Anyway, if you meant `sec^2x`, here it is:

`[1]" "sec^2x-csc^2x`

Pythagorean Identity: `sec^2alpha=1+tan^2alpha`

`[2]" "-=(1+tan^2x)-csc^2x`

Pythagorean Identity: `csc^2alpha=1+cot^2alpha`

`[3]" "-=(1+tan^2x)-(1+cot^2x)`

`[4]" "=1+tan^2x-1-cot^2x`

`[5]" "=tan^2x-cot^2x`

`color(blue)( :.sec^2x-csc^2x=tan^2x-cot^2x)`