How do you verify the identity (sec^2x-1)cos^2x=sin^2x?

Oct 4, 2016

Simplify the left side of the identity without changing the right side of the identity at all. The left side will simplify to ${\sin}^{2} x$.

Explanation:

$\left({\sec}^{2} x - 1\right) {\cos}^{2} x = {\sin}^{2} x$
Distribute ${\cos}^{2} x$:
${\sec}^{2} x {\cos}^{2} x - {\cos}^{2} x = {\sin}^{2} x$
Recall that ${\sec}^{2} x$ is defined to be the reciprocal of ${\cos}^{2} x$, or $\frac{1}{\cos} ^ 2 x$. Therefore, we now have:
${\cos}^{2} \frac{x}{\cos} ^ 2 x - {\cos}^{2} x = {\sin}^{2} x$
This simplifies to:
$1 - {\cos}^{2} x = {\sin}^{2} x$
Now recall that ${\sin}^{2} x + {\cos}^{2} x = 1$, which can be transformed using the Subtraction Property of Equality to ${\sin}^{2} x = 1 - {\cos}^{2} x$. Substituting will then give us:
${\sin}^{2} x = {\sin}^{2} x$
So therefore, the identity has been verified.