Question

How do you prove sin3θ = 3cos^2θ sinθ -sin^3θ ?

Answer 1

Please see below.

Explanation:

We have to prove :

`sin3θ = 3cos^2θ sinθ -sin^3θ`

We know that ,

`color(brown)((1)sin(x+y)=sinxcosy+cosxsiny`

Let ,

`LHS=sin3theta`

`LHS=sin(2theta+theta)to[use, (1)]`

`LHS=color(red)(sin2theta)costheta+color(blue)(cos2theta)sintheta`

`LHS=color(red)(2sinthetacostheta)costheta+color(blue)((cos^2theta-sin^2theta))sintheta`

`LHS=2cos^2thetasintheta+cos^2thetasintheta-sin^3theta`

`LHS=3cos^2thetasintheta-sin^3theta`

`LHS=RHS`