How do you express cot^3theta+tan^3theta in terms of non-exponential trigonometric functions?

1 Answer
Oct 15, 2016

cot^3theta+tan^3theta

=cos^3theta/sin^3theta+sin^3theta/cos^3theta

=(cos^6theta+sin^6theta)/(sin^3thetacos^3theta)

=((cos^2theta+sin^2theta)^3-3(cos^2thetasin^2theta)(cos^2theta+sin^2theta))/(sin^3thetacos^3theta)

=(1-3(cos^2thetasin^2theta)*1)/(sin^3thetacos^3theta)

=(8-24(cos^2thetasin^2theta))/(2^3sin^3thetacos^3theta)

=(8-6(2costhetasintheta)^2)/(2sinthetacostheta)^3

=(8-3*2sin^2 2theta)/(sin^3 2theta)

=(8-3*(1-cos4theta))/(1/4(3sin2theta-sin6theta)

=(5+3cos4theta)/(1/4(3sin2theta-sin6theta)

=(20+12cos4theta)/(3sin2theta-sin6theta)