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

1 Answer
Mar 5, 2018

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