Prove

Question

sin^5(x)=(1-2cos^2 x +cos^4 x)(sinx)?

Answer 1

Left hand side identity = sin⁵(x)

= sin(x) sin⁴(x)

= sin(x) [sin²(x)]²

{Apply Pythagorean identities sin²x + cos²x = 1}

= sin(x) [1 - cos²(x)]²

= sin(x){1 + [cos²(x)]² - 2cos²(x)}

= sin(x)[1 + cos⁴(x) - 2cos²(x)]

= [1 - 2cos²(x) + cos⁴(x) ][sin(x)]

= Right hand side identity.