What is sintheta-tantheta+cotthetasinθ−tanθ+cotθ in terms of costhetacosθ?

1 Answer
Narad T.
Apr 8, 2018

Please see the explanation below.

Explanation:

We need

sin^2theta+cos^2theta=1sin2θ+cos2θ=1

tantheta=sintheta/costhetatanθ=sinθcosθ

cottheta=costheta/sinthetacotθ=cosθsinθ

Therefore,

sintheta-tantheta+cotthetasinθ−tanθ+cotθ

=sintheta-sintheta/costheta+costheta/sintheta=sinθ−sinθcosθ+cosθsinθ

=sqrt(1-cos^2theta)-sqrt(1-cos^2theta)/costheta+costheta/sqrt(1-cos^2theta)=√1−cos2θ−√1−cos2θcosθ+cosθ√1−cos2θ

=(costheta(1-cos^2theta)-(1-cos^2theta)+cos^2theta)/(costhetasqrt(1-cos^2theta))=cosθ(1−cos2θ)−(1−cos2θ)+cos2θcosθ√1−cos2θ

=(costheta-cos^3theta-1+cos^2theta+cos^2theta)/(costhetasqrt(1-cos^2theta))=cosθ−cos3θ−1+cos2θ+cos2θcosθ√1−cos2θ

=(costheta+2cos^2theta-cos^3theta-1)/(costhetasqrt(1-cos^2theta))=cosθ+2cos2θ−cos3θ−1cosθ√1−cos2θ

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