using a substitution, like u = 2x, may cut down the work some.
my first tipp would be that you express the equation only with the term sin(2x) and cos(2x).
cos(10x) = cos(8x + 2x) = cos(8x)sin(2x) - sin(8x)cos(2x)
and again...and again
and finally you can substitution cos(2x) = y and sin(2x) = sqrt(1-y^2)
So you get an algebraic equation. The problem is now the too high degree of the polinom but maybe you get a special polinom which is solvable, i dunno.
Normally, we are not allowed to work on both sides of the identity.
But I cannot find a way to transform one side into the other.
We can use two Sum-to-Product Identities:
The left side is: .
The right side is: .
The two sides are equal, but it is not a satisfactory proof of an identity.