Carla Murphy

## Answered question

2021-12-26

Prove $-4{\mathrm{sin}}^{4}\left(\frac{12}{x}\right)=-4{\mathrm{sin}}^{2}\left(\frac{12}{x}\right)+{\mathrm{sin}}^{2}x$
I want to prove :
$-4{\mathrm{sin}}^{4}\left(\frac{x}{2}\right)=-4{\mathrm{sin}}^{2}\left(\frac{x}{2}\right)+{\mathrm{sin}}^{2}x$
I try to factoring :
$-4{\mathrm{sin}}^{2}\left(\frac{x}{2}\right)+{\mathrm{sin}}^{2}x=\left(\mathrm{sin}x+2\mathrm{sin}\left(\frac{x}{2}\right)\right)\left(\mathrm{sin}x-2\mathrm{sin}\left(\frac{x}{2}\right)\right)$
but I can't simplify it to left hand side.

### Answer & Explanation

sonorous9n

Beginner2021-12-27Added 34 answers

Consider, right hand side
$-4{\mathrm{sin}}^{2}\left(\frac{x}{2}\right)+{\mathrm{sin}}^{2}x=-4{\mathrm{sin}}^{2}\left(\frac{x}{2}\right)+{\left(2\mathrm{sin}\left(\frac{x}{2}\right)\mathrm{cos}\left(\frac{x}{2}\right)\right)}^{2}$
$=-4{\mathrm{sin}}^{2}\left(\frac{x}{2}\right)+4{\mathrm{sin}}^{2}\left(\frac{x}{2}\right){\mathrm{cos}}^{2}\left(\frac{x}{2}\right)$
$=-4{\mathrm{sin}}^{2}\left(\frac{x}{2}\right)\left(1-{\mathrm{cos}}^{2}\left(\frac{x}{2}\right)\right)$
$=-4{\mathrm{sin}}^{2}\left(\frac{x}{2}\right){\mathrm{sin}}^{2}\left(\frac{x}{2}\right)$
$=-4{\mathrm{sin}}^{4}\left(\frac{x}{2}\right)$

Piosellisf

Beginner2021-12-28Added 40 answers

Hint:
$\mathrm{sin}x=2\mathrm{sin}\left(\frac{x}{2}\right)\mathrm{cos}\left(\frac{x}{2}\right)$

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