# How do you use the fundamental identities to prove other identities?

Divide the fundamental identity ${\sin}^{2} x + {\cos}^{2} x = 1$ by ${\sin}^{2} x$ or ${\cos}^{2} x$ to derive the other two:
${\sin}^{2} \frac{x}{\sin} ^ 2 x + {\cos}^{2} \frac{x}{\sin} ^ 2 x = \frac{1}{\sin} ^ 2 x$
$1 + {\cot}^{2} x = {\csc}^{2} x$
${\sin}^{2} \frac{x}{\cos} ^ 2 x + {\cos}^{2} \frac{x}{\cos} ^ 2 x = \frac{1}{\cos} ^ 2 x$
${\tan}^{2} x + 1 = {\sec}^{2} x$