# How do you simplify cos(pi/2 - x) * csc(-x)?

Apr 5, 2018

$\cos \left(\frac{\pi}{2} - x\right) \cdot \csc \left(- x\right) = - 1$ for $x$ not a multiple of $\pi$.

#### Explanation:

We know that the angles of a right triangle must sum to $\pi$ radians. Since the right angle is $\frac{\pi}{2}$ radians, the other two angles must sum to $\frac{\pi}{2}$ radians. Therefore $\frac{\pi}{2}$ minus the adjacent angle must be the opposite angle. Another way of saying this is $\cos \left(\frac{\pi}{2} - x\right) = \sin x$, so

$\cos \left(\frac{\pi}{2} - x\right) \cdot \csc \left(- x\right) = \sin x \cdot \csc \left(- x\right)$.

The definition of $\csc x$ says that $\csc x = \frac{1}{\sin} x$, so

$\sin x \cdot \csc \left(- x\right) = \sin \frac{x}{\sin} \left(- x\right)$

We know that $\sin x$ is an odd function so $\sin \left(- x\right) = - \sin x$ and we can write

$\sin \frac{x}{\sin} \left(- x\right) = - \sin \frac{x}{\sin} x = - 1$.

However, this breaks down when $x$ is a multiple of $\pi$ because $\csc \left(n \pi\right)$ is not defined.