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.