Question

I need to get the left side to equal the right side. Keeping the right side alone and not changing it.

Answer 1

To prove the trigonometric equation:

$\sin ^{3} x+\cos ^{3} x=(\sin x+\cos x)(1-\sin x \cos x)$

$RHS=(\sin x+\cos x)(1-\sin x \cos x)$

We know that $\sin^2x +\cos^2x=1$, substitute this in place of 1.

       $=(\sin x+\cos x)(\sin^2x +\cos^2x-\sin x \cos x)$

Multiply each term of the first term with each term of the 2nd term.

       $=\sin^3x + \sin x \cos^2x-\sin^2 x \cos x+\cos x \sin^2 x + \cos^3 x-\sin x\cos^2 x$

Group like terms together.

       $=\sin^3x +( \sin x \cos^2x-\sin x\cos^2 x)+(\cos x \sin^2 x-\sin^2 x \cos x) + \cos^3 x$

       $=\sin^3x +( 0)+(0) + \cos^3 x$

       $=\sin^3x + \cos^3 x$

       = LHS

RHS = LHS

$\sin ^{3} x+\cos ^{3} x=(\sin x+\cos x)(1-\sin x \cos x)$

Hence proved.