Question

How to prove that cos^2x-sin^2x/sinxcosx=2-sec^2x/tanx?

Answer 1

See explanation

Explanation:

We want to verify the identity

`(cos^2(x)-sin^2(x))/(cos(x)sin(x))=(2-sec^2(x))/tan(x)`

We will use the pythagorean trigonometric identity

• `cos^2(x)+sin^2(x)=1`

`RHS=(2-sec^2(x))/tan(x)`

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

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

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

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

`=(cos^2(x)-sin^2(x))/(cos(x)sin(x))=LHS`