# How do you verify (cosX+sinX)/(cscX+secX) = (cosX)(sinX)?

Feb 22, 2016

Using the definitions of secant and cosecant

• $\csc \left(x\right) = \frac{1}{\sin} \left(x\right)$
• $\sec \left(x\right) = \frac{1}{\cos} \left(x\right)$

We have

$\frac{\cos \left(x\right) + \sin \left(x\right)}{\csc \left(x\right) + \sec \left(x\right)} = \frac{\cos \left(x\right) + \sin \left(x\right)}{\frac{1}{\sin} \left(x\right) + \frac{1}{\cos} \left(x\right)}$

$= \frac{\cos \left(x\right) \sin \left(x\right)}{\cos \left(x\right) \sin \left(x\right)} \cdot \frac{\cos \left(x\right) + \sin \left(x\right)}{\frac{1}{\sin} \left(x\right) + \frac{1}{\cos} \left(x\right)}$

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

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

$= \cos \left(x\right) \sin \left(x\right)$

(Note that this identity is only true where secant and cosecant are defined, that is, where $\sin \left(x\right) \ne 0$ and $\cos \left(x\right) \ne 0$)