# How can I prove the following equation is an identity? 1+sec^2(x)sin^2(x)=sec^2(x)

Mar 18, 2018

$1 + {\sec}^{2} \left(x\right) {\sin}^{2} \left(x\right) = {\sec}^{2} \left(x\right)$

Use reciprocal identity:
${\sec}^{2} x = \frac{1}{\cos} ^ 2 x$

Therefore:
$1 + \frac{1}{\cos} ^ 2 x \cdot {\sin}^{2} \left(x\right) = {\sec}^{2} \left(x\right)$
$1 + {\sin}^{2} \frac{x}{\cos} ^ 2 x = {\sec}^{2} \left(x\right)$

Use the quotient identity:
${\sin}^{2} \frac{x}{\cos} ^ 2 x = {\tan}^{2} x$

Therefore:
$1 + {\tan}^{2} x = {\sec}^{2} \left(x\right)$

Use the Pythagorean identity:
$1 + {\tan}^{2} x = {\sec}^{2} \left(x\right)$

Therefore:
${\sec}^{2} x = {\sec}^{2} x$