How do you verify the identity tanthetacsc^2theta-tantheta=cotthetatanθcsc2θtanθ=cotθ?

2 Answers
Sep 10, 2016

Proof below

Explanation:

tantheta * csc^2theta - tanthetatanθcsc2θtanθ
=sintheta/costheta * (1/sintheta)^2 - sintheta/costheta=sinθcosθ(1sinθ)2sinθcosθ
=sintheta/costheta * 1/sin^2theta - sintheta/costheta=sinθcosθ1sin2θsinθcosθ
=1/(sinthetacostheta) - sintheta/costheta=1sinθcosθsinθcosθ
=(1-sin^2theta)/(sinthetacostheta)=1sin2θsinθcosθ
=cos^2theta/(sinthetacostheta)=cos2θsinθcosθ
=costheta/sintheta=cosθsinθ
=cottheta=cotθ

Note that sin^2theta + cos^2theta = 1sin2θ+cos2θ=1, therefore cos^2theta = 1- sin^2thetacos2θ=1sin2θ

Sep 10, 2016

LHS=tantheta * csc^2theta - tanthetaLHS=tanθcsc2θtanθ

=tantheta ( csc^2theta - 1)=tanθ(csc2θ1)

=tantheta ( 1+cot^2theta - 1)=tanθ(1+cot2θ1)

=tantheta * cot^2theta=tanθcot2θ

=cottheta=RHS=cotθ=RHS