Use the sum-to-product identities to rewrite the following expression in terms containing only first powers of tangent. sin8x−sin2xcos8x+cos2x

The given expression is sin8x - sin2x*cos8x + cos2x.

First, we need to recall the sum-to-product identities:

sin(A)cos(B) = 1/2[sin(A-B) + sin(A+B)] cos(A)cos(B) = 1/2[cos(A-B) + cos(A+B)] sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)]

Now, let's rewrite the given expression using these identities:

sin8x - 1/2[sin(8x-2x) + sin(8x+2x)] + 1/2[cos(2x-2x) + cos(2x+2x)]

This simplifies to:

sin8x - 1/2[sin6x + sin10x] + 1/2[cos0 + cos4x]

Now, we know that cos0 = 1, so the expression becomes:

sin8x - 1/2sin6x - 1/2sin10x + 1/2 + 1/2cos4x

This is the expression rewritten in terms of first powers of tangent.