Please help me with this identity: (cosx+sinx)(1-sinxcosx)=cos ^{3}x+sin ^{3}x

Question

anonymous · Answer

(cosx+sinx)(1-sinxcosx)=cos ^{3}x+sin ^{3}x

anonymous · Answer

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

zehanz · Answer

You will need this factorisation: a³+b³ = (a+b)(a²-ab+b²).
You can verify it by doing a long division of a³+b³ by a+b.

Because this is a trig problem, I don't think you are required to do that here, so:

$\cos^3x+\sin^3x=\(\cos x+\sin x)(\cos^2x - \sin x\cos x+ \sin^2x)$.

If you compare this reworked right-hand side with the LHS, I think you are almost done...

zehanz · Answer

(ignore the red "\" above...)

zehanz · Answer

Maybe it could also be done without first looking at the RHS.
If you replace 1 with sin²x+cos²x in the LHS, if you work out all the multiplications (6 in total) you should eventually arrive at the RHS...

anonymous · Answer

thanks, yeah i tried multiplying out the left side but didn't replace the one and got stuck.  Using the sum of 2 cubes trick got me to the right answer.  Thank you for all your help today

zehanz · Answer

No problem!