# Question #13e99

Oct 28, 2017

see explanation...

#### Explanation:

First, multiply out in the numerator:

${\left(\sin \left(t\right) + \cos \left(t\right)\right)}^{2} / \left(\sin \left(t\right) \cos \left(t\right)\right) = \frac{{\sin}^{2} \left(t\right) + 2 \sin \left(t\right) \cos \left(t\right) + {\cos}^{2} \left(t\right)}{\sin \left(t\right) \cos \left(t\right)}$

Re-order the terms in the numerator a bit, and use the fact that ${\sin}^{2} \left(t\right) + {\cos}^{2} \left(t\right) = 1$, giving:

$\frac{1 + 2 \sin \left(t\right) \cos \left(t\right)}{\sin \left(t\right) \cos \left(t\right)}$
$= \frac{1}{\left(\sin \left(t\right) \cos \left(t\right)\right)} + \frac{2 \left(\sin \left(t\right) \cos \left(t\right)\right)}{\sin \left(t\right) \cos \left(t\right)}$

...you can now factor/separate the first term. Also, note that in the right term, the $\sin \left(t\right) \cos \left(t\right)$ in numerator and denominator cancels out. So you have:

$\left(\frac{1}{\sin} \left(t\right)\right) \cdot \left(\frac{1}{\cos} \left(t\right)\right) + 2$

...now remember $\frac{1}{\cos} \left(t\right) = \sec \left(t\right)$, and $\frac{1}{\sin} \left(t\right) = \csc \left(t\right)$, so it all works out to:

$\csc \left(t\right) \sec \left(t\right) + 2$

GOOD LUCK