# How do I prove this equation is an identity? Sin(x)/csc(x)+Cos(x)/Sec(x)=1

Mar 22, 2018

Below.

#### Explanation:

$\csc \left(x\right) = \frac{1}{\sin} \left(x\right)$
$\sec \left(x\right) = \frac{1}{\cos} \left(x\right)$
$\therefore \sin \frac{x}{\csc} \left(x\right) = \sin \frac{x}{\frac{1}{\sin} \left(x\right)} = {\sin}^{2} \left(x\right)$

Similarly,

$\cos \frac{x}{\sec} \left(x\right) = \cos \frac{x}{\frac{1}{\sec} \left(x\right)} = {\cos}^{2} \left(x\right)$

Therefore,

$L H S = \sin \frac{x}{\csc} \left(x\right) + \cos \frac{x}{\sec} \left(x\right) = {\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right) = 1 = R H S$