# What is the equivalent to sec[arcsin(x-1)] in terms of an algebraic expression?

## $\sec \left[\arcsin \left(x - 1\right)\right]$

Apr 9, 2018

#### Explanation:

Let $\theta = \arcsin \left(x - 1\right)$ such that $\theta$ is the angle between $c$ and $a$ in the triangle above.

This means that $\sin \left(\theta\right) = x - 1$

Hence, $b = x - 1$ and $c = 1$

Using pythagoras shows that $a = \sqrt{1 - {\left(x - 1\right)}^{2}}$

Now it can be easily seen that

$\sec \left(\theta\right) = \frac{1}{\sqrt{1 - {\left(x - 1\right)}^{2}}}$

Therefore
$\sec \left[\arcsin \left(x - 1\right)\right] = \frac{1}{\sqrt{1 - {\left(x - 1\right)}^{2}}}$