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

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Answer 1 · 2018-03-05T07:55:00

See 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

#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#