# How do you show that (costheta)(sectheta) = 1 if theta=pi/4?

Feb 3, 2015

Let's look at the triangle:

We'll use $\angle A$ in stead of $\theta$ because of the picture.

Now $\cos A = b / h$ and $\sec A = h / b$ by definition.

So their product is:

$\cos A \cdot \sec A = \frac{b}{h} \cdot \frac{h}{b} = 1$ for all values of $A$

If we come closer to $A = \frac{\pi}{4}$ the value of
$\cos A \to 0$ while the value of $\sec A \to \infty$

As long as $A$ is a little bit smaller than $\frac{\pi}{4}$ then $\cos A \cdot \sec A$ still come to $1$. We can get as close as we want, the answer is still $1$.

Summary
${\lim}_{A \to \pi / 4} \cos A \cdot \sec A = 1$