Prove that tan^2(x) + 1 = sec^2(x)

tan^2(x) + 1 = sec^2(x)

The function

\tan^2(x) + 1

is equal to

\sec^2(x)

.

Proof. We have seen here that:

\begin{align*} \sin^2(x) + \cos^2(x) = 1. \end{align*}

Now multiply both sides with

\frac{1}{\cos^2(x)}

gives us:

\begin{align*} \frac{\sin^2(x)}{\cos^2(x)} + 1 = \frac{1}{\cos^2(x)}. \end{align*}

We do know that

\frac{\sin^2(x)}{\cos^2(x)} = \tan^2(x)

and

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

. Therefore, we get the following:

\begin{align*} \tan^2(x) + 1 = \sec^2(x). \end{align*}

Therefore,

\tan^2(x) + 1

is equal to

\sec^2(x)