Gregory Jones

2022-01-03

How can I prove that:
${\left(1+\mathrm{sin}x+\mathrm{cos}x\right)}^{2}=2\left(1+\mathrm{cos}x\right)\left(1+\mathrm{sin}x\right)$
Ive

stomachdm

To achieve your goal, you want to factor out $\left(1+\mathrm{sin}x\right)$:
${\left(1+\mathrm{sin}x\right)}^{2}+{\mathrm{cos}}^{2}x+2\left(1+\mathrm{sin}x\right)\mathrm{cos}x=$
${\left(1+\mathrm{sin}x\right)}^{2}+\left(1-{\mathrm{sin}}^{2}x\right)+2\left(1+\mathrm{sin}x\right)\mathrm{cos}x=$
$\left[\left(1+\mathrm{sin}x\right)+\left(1-\mathrm{sin}x\right)+2\mathrm{cos}x\right]\left(1+\mathrm{sin}x\right)=$
$\left[2+2\mathrm{cos}x\right]\left(1+\mathrm{sin}x\right)=2\left(1+\mathrm{cos}x\right)\left(1+\mathrm{sin}x\right)$

macalpinee3

Just continue expanding like you have, to obtain
$1+{\mathrm{sin}}^{2}\left(x\right)+2\mathrm{sin}\left(x\right)+{\mathrm{cos}}^{2}\left(x\right)+2\mathrm{cos}\left(x\right)+2\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)$
which then beomes
$2+2\mathrm{sin}\left(x\right)+2\mathrm{cos}\left(x\right)+2\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)$
From there, factor out the two and notice that 1+a+b+ab=(1+a)(1+b).

Vasquez

$\left(1+\mathrm{sin}x+\mathrm{cos}x{\right)}^{2}=1+{\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x+2\mathrm{sin}x+2\mathrm{cos}x+2\mathrm{sin}x\mathrm{cos}x$
$=2+2\mathrm{sin}x+2\mathrm{cos}x+2\mathrm{sin}x\mathrm{cos}x$
$=2\left(1+\mathrm{sin}x\right)\left(1+\mathrm{cos}x\right)$

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