# How do you prove 1 + sin² x = 1/sec² x?

##### 1 Answer
Jul 5, 2015

The given equation is not true (except for values of $x$ equal to integer multiples of $\pi$).

#### Explanation:

$\sec \left(x\right) = \frac{1}{\cos} \left(x\right)$

The maximum value of $\left\mid \cos \left(x\right) \right\mid$ is 1

$\Rightarrow$ the minimum value of $\left\mid \sec \left(x\right) \right\mid$ is 1
and
the maximum value of $\frac{1}{{\sec}^{2} \left(x\right)}$ is 1

${\sin}^{2} \left(x\right)$ has a minimum value of 0
$1 + {\sin}^{2} \left(x\right)$ has a minimum value of 1

The only time $1 + {\sin}^{2} \left(x\right) = \frac{1}{{\sec}^{2} \left(x\right)}$
is when
$\textcolor{w h i t e}{\text{XXXX}}$${\sec}^{2} \left(x\right) = 1$ and ${\sin}^{2} \left(x\right) = 1$

These conditions are only true for $x = 0$ and any other value of $x = k \pi$ (with $k \epsilon \mathbb{Z}$).