How do you use sum and difference identities to simplify csc(pi/2 +x) - sec(pi - x)?

Mar 14, 2018

0

Explanation:

$\csc \left(\theta\right) = \frac{1}{\sin} \left(\theta\right) , \sec \left(\theta\right) = \frac{1}{\cos} \left(\theta\right)$
$\frac{1}{\sin \left(\frac{\pi}{2} + x\right)} - \frac{1}{\cos \left(\pi - x\right)}$

$\sin \left(a + b\right) = \sin a \cdot \cos b + \cos a \cdot \sin b$
$\cos \left(a - b\right) = \cos a \cdot \cos b + \sin a \cdot \sin b$

$\frac{1}{\sin \left(\frac{\pi}{2}\right) \cdot \cos x + \cos \left(\frac{\pi}{2}\right) \cdot \sin x} - \frac{1}{\cos \pi \cdot \cos x + \sin \pi \cdot \sin x}$
Using either calculator or unit circle,
$\cos \left(\frac{\pi}{2}\right) = 0 , \sin \left(\pi\right) = 0 , \cos \left(\pi\right) = 1 , \sin \left(\frac{\pi}{2}\right) = 1$

$\frac{1}{\cos x} - \frac{1}{\cos x} = 0$