sin6x−sin4x / cos6x+cos4x

Question

Use the sum-to-product identities to rewrite the following expression in terms containing only first powers of tangent.

Answer 1

Using, sin(A)-sin(B) = 2cos((A+B)/2)sin((A-B)/2) & cos(A)+cos(B) = 2cos((A+B)/2)cos((A-B)/2)

sin(6x)-sin(4x)/cos(6x)+cos(4x) = 2cos((6x+4x)/2)sin((6x-4x)/2) / 2cos((6x+4x)/2)cos((6x-4x)/2) =

2cos((10x)/2)sin((2x)/2) / 2cos((10x)/2)cos((2x)/2) = sin(2x/2) / cos(2x/2) = tan (2x/2) = tan (x)