Prove sin3x4sinxcosxsinx Prove sin3x4sinxcosxsinxSolutionsin

Solution

sin3x=4sinxcos^x-sinx

LHS sin(3x) = sin(2x +x)

use sin(A+B) = sinAcosB +cosAsinB

sin(3x) = sin(2x +x) = sin2xcosx +cos2xsinx

Use trig. formula : sin2x = 2sinxcosx ; cos2x = cos^2x -sin^2x

sin(3x) = sin(2x +x) = sin2xcosx +cos2xsinx

= 2sinxcosx^2x + (cos^2x -sin^2x)sinx

= 2sinxcos^2x + sinxcos^2x - sin^3x

=3sinxcos^2x -sin^3x

=3sinx(1-sin^2x) -sin^3x

= 3sinx-3sin^3x -sin^3x

=3sinx -4sin^3x

=3sinx -4sin^2xsinx = 3sinx -4(1-cos^2x)sinx

= 4cos^2xsinx -sinx

= RHS