How do you express sin^4theta+cot^2theta -cos^4 theta in terms of non-exponential trigonometric functions?

1 Answer
Feb 1, 2016

(sintheta+costheta)(sintheta-costheta)+(csctheta+1)(csctheta-1)

Explanation:

Rewrite to group the sine and cosine terms.

=sin^4theta-cos^4theta+cot^2theta

Simplify the first two terms as a difference of squares.

=(sin^2theta+cos^2theta)(sin^2theta-cos^2theta)+cot^2theta

Note that sin^2theta+cos^2theta=1 through the Pythagorean Identity.

=sin^2theta-cos^2theta+cot^2theta

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

=(sintheta+costheta)(sintheta-costheta)+cot^2theta

Use the identity: cot^2theta+1=csc^2theta to say that cot^2theta=csc^2theta-1.

=(sintheta+costheta)(sintheta-costheta)+csc^2theta-1

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

=(sintheta+costheta)(sintheta-costheta)+(csctheta+1)(csctheta-1)