# How do you prove sin x + cos x * cot x = csc x?

Nov 30, 2016

We will use the following identities to attack the problem:

$\cot x = \frac{1}{\tan} x = \frac{1}{\sin \frac{x}{\cos} x} = \cos \frac{x}{\sin} x$
$\csc x = \frac{1}{\sin} x$
${\cos}^{2} x + {\sin}^{2} x = 1$

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$\sin x + \cos x \cdot \cos \frac{x}{\sin} x = \frac{1}{\sin} x$

$\sin x + {\cos}^{2} \frac{x}{\sin} x = \frac{1}{\sin} x$

Put the left hand side on a common denominator.

${\sin}^{2} \frac{x}{\sin} x + {\cos}^{2} \frac{x}{\sin} x = \frac{1}{\sin} x$

$\frac{{\sin}^{2} x + {\cos}^{2} x}{\sin} x = \frac{1}{\sin} x$

$\frac{1}{\sin} x = \frac{1}{\sin} x$

$L H S = R H S$

Identity proved!

Hopefully this helps!