How do you simplify f(theta)=csc2theta-sec2theta-3cot2thetaf(θ)=csc2θ−sec2θ−3cot2θ to trigonometric functions of a unit thetaθ?

1 Answer
Kevin D.
Oct 14, 2017

f(theta)=1/(sin(2theta))=1/(2sinthetacostheta)-1/(2cos^2theta-1)-(2-1tan^2theta)/(2tantheta)f(θ)=1sin(2θ)=12sinθcosθ−12cos2θ−1−2−1tan2θ2tanθ

Explanation:

Break the function down by term:
csc(2theta)=1/(sin(2theta))=1/(2sinthetacostheta)csc(2θ)=1sin(2θ)=12sinθcosθ

sec(2theta)=1/(cos(2theta))=1/(cos^2theta-sin^2theta)=1/(2cos^2theta-1)sec(2θ)=1cos(2θ)=1cos2θ−sin2θ=12cos2θ−1

cot(2theta)=1/tan(2theta)=(1-tan^2theta)/(2tantheta)cot(2θ)=1tan(2θ)=1−tan2θ2tanθ

Put this hell together:

f(theta)=1/(sin(2theta))=1/(2sinthetacostheta)-1/(2cos^2theta-1)-(3-3tan^2theta)/(2tantheta)f(θ)=1sin(2θ)=12sinθcosθ−12cos2θ−1−3−3tan2θ2tanθ

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