Verify the identity Verify the identity sin5 x cos2 x cos2

Verify the identity

Verify the identity sin^5 x cos^2 x = (cos^2 x - 2cos^4 x + cos^6 x) sin x

Solution

Left side : sin⁵x cos²x

= sin x sin⁴x cos²x

= sin x (1-cos²x)² cos²x [Since sin²x=1-cos²x]

= sin x ( 1 + cos⁴x - 2cos²x ) cos²x [ Since (1-cos²x)^{2 =} (1-cos²x)(1-cos²x) = 1 + cos⁴x - 2cos²x ]

= sin x (cos²x + cos⁶x - 2cos⁴x) [Distributing cos²x inside parenthesis]

= Right Side

Hence, identity verified.