question

sin²x=a
cos²x=b
a+b=1

so LHS is a⁵+b⁵ = (a+b) (a⁴-a³b+a²b²-ab³+b⁴)

=a⁴-a³b+a²b²-ab³+b⁴

cos⁴(2x) = (b-a)⁴

also a⁴+b⁴ = a⁴+b⁴ -2a²b²+2a²b²
= (a²-b²)²+2a²b²
= (a-b)²(a+b)²+2a²b²
= (a-b)²+2a²b²
= a²+b²-2ab+2a²b²

also make similar reductions for cubes!

a³b+ab³ = ab(a²+b²)

ab(a²+b²+2ab-2ab)
= ab(1-2ab)

using these u will be able to reduce everything in terms of ab and (ab)² wchich is a quadratic...

then it should be fairly simple :)