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

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Answer 1 · 2018-08-12T16:31:39

Please see below.

We have to prove :

$sin3θ = 3cos^2θ sinθ -sin^3θ$

We know that ,

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

Let ,

$LHS=sin3theta$

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

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

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

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

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

$LHS=RHS$