# Verify the Identity? Sinx / 1-cos^2x = cscx

Feb 23, 2018

To solve this, we need to use the Pythagorean Trig Identity.

#### Explanation:

The Pythagorean Identity states:

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

We manipulate this to get either ${\cos}^{2} x$ or ${\sin}^{2} x$ by itself. For this problem, we want ${\sin}^{2} x$ by itself. To do this, we can simply subtract the ${\cos}^{2} x$ over to the other side, making it:

${\sin}^{2} x = 1 - {\cos}^{2} x$

Knowing this, we can verify the trigonometric equation. Since we now know that $1 - {\cos}^{2} x$ equals ${\sin}^{2} x$, we can substitute that in, giving us:

$\sin \frac{x}{\sin} ^ 2 x = \csc x$

From here, we can do a simple cancellation of a $\sin x$ in the numerator and denominator, which leaves us with:

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

And then:

$\csc x = \csc x$