Question

Sin^5xcos^2x=(cos^2x-2cos^4x+cos^6x)sinx

Please show steps on how to solve this I'm so confused!!! Thanks!!!

Answer 1

`sin^5xcos^2x = (cos^2x-2cos^4x+cos^6x)sinx\ is\ proved`

Solution:

Given that we have to prove:

`sin^5xcos^2x = (cos^2x-2cos^4x+cos^6x)sinx`

Let us first take the left hand side of equation

`sin^5xcos^2x\\\\Split\ sin^5x\ as\ (sin^4x)(sinx)`

`(sin^4x)(sinx)(cos^2x)\\\\Which\ is\\\\(sinx)(sin^4x)(cos^2x) ------ eqn 1`

Now take the right side of equation

`(cos^2x-2cos^4x+cos^6x)sinx\\\\Take\ cos^2x\ as\ common\ term\ out\\\\(1-2cos^2x + cos^4x)cos^2x\ sinx`

`By\ algebraic\ identity,\\\\(a-b)^2 = a^2-2ab + b^2`

`Therefore,\\\\(1-cos^2x)^2cos^2x\ sinx\\\\We\ know\ that\\\\1-cos^2x = sin^2x\\\\Therefore\\\\sin^4xcos^2x sinx ----- eqn 2`

Eqn 1 = eqn 2

Therefore, L.H.S = R.H.S

`sin^5xcos^2x = (cos^2x-2cos^4x+cos^6x)sinx`

Thus proved