(Sin^3x-cos^3x)/(sinx-cosx) – cosx/sqrt(1+cot^2x)-2tanxcotx=-1 where x∈(0,2pi)

Question

general value of x.

Answer 1

sin^3 - cos^3 = (sin-cos)(sin^2 + sin*cos + cos^2) = (sin-cos)(1+sin*cos)

so, (sin^3-cos^3)/(sin-cos) = 1+sin*cos

1+cot^2 = csc^2, so
cos/csc = sin*cos

tan*cot=1, so we have

1+sin*cos - sin*cos - 2 = -1

which is true for all values of x