How do you express sin4θ+cot2θcos4θ in terms of non-exponential trigonometric functions?

1 Answer
Feb 1, 2016

(sinθ+cosθ)(sinθcosθ)+(cscθ+1)(cscθ1)

Explanation:

Rewrite to group the sine and cosine terms.

=sin4θcos4θ+cot2θ

Simplify the first two terms as a difference of squares.

=(sin2θ+cos2θ)(sin2θcos2θ)+cot2θ

Note that sin2θ+cos2θ=1 through the Pythagorean Identity.

=sin2θcos2θ+cot2θ

The first two terms can again be factored as a difference of squares.

=(sinθ+cosθ)(sinθcosθ)+cot2θ

Use the identity: cot2θ+1=csc2θ to say that cot2θ=csc2θ1.

=(sinθ+cosθ)(sinθcosθ)+csc2θ1

Again, the last two terms can be factored as a difference of squares.

=(sinθ+cosθ)(sinθcosθ)+(cscθ+1)(cscθ1)