# How do you simplify sec xcos (frac{\pi}{2} - x )?

Dec 15, 2014

The application of co-functions will make it easy to simplify this expression.

However, I am going to present how the co-function is derived so you can have a better understanding as to why it came as such.

The co-function of $\cos \left(\frac{\pi}{2}\right) - x$ is $\sin x$

So how did this happen?

Examine the following equation

$\cos \left(A - B\right) = \cos A \cdot \cos B + \sin A \cdot \sin B$

Thus, applying this formula to $\cos \left(\frac{\pi}{2}\right) - x$

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

$\cos \left(\frac{\pi}{2}\right) = 0 \mathmr{and} \sin \left(\frac{\pi}{2}\right) = 1$

therefore,

$\cos \left[\left(\frac{\pi}{2}\right) - x\right] = \sin x$

Thus,

$\sec x \cdot \sin x$

but,

$\sec x = \frac{1}{\cos} x$

then

$\frac{1}{\cos} x \cdot \sin x$

$\tan x$