Gavin Frye

## Answered question

2022-01-27

Show that when written in terms of t, where $t=\mathrm{tan}\left(\frac{x}{2}\right)$ the expression $2\left(1+\mathrm{cos}\left(x\right)\right)\left(5\mathrm{sin}\left(x\right)+12\mathrm{cos}\left(x\right)+13\right)$ is a perfect square.

### Answer & Explanation

rakije2v

Beginner2022-01-28Added 12 answers

$2\left(1+\mathrm{cos}\left(x\right)\right)\left(5\mathrm{sin}\left(x\right)+12\mathrm{cos}\left(x\right)+13\right)=2\left(1+\frac{1-{t}^{2}}{1+{t}^{2}}\right)\left(\frac{10t}{1+{t}^{2}}+\frac{12-12{t}^{2}}{1+{t}^{2}}+13\right)$
$=\frac{4}{{\left(1+{t}^{2}\right)}^{2}}{\left(t+5\right)}^{2}$

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