# How do you simplify (sina+tana)/(1+cosa)?

Oct 11, 2016

$\frac{\sin \left(a\right) + \tan \left(a\right)}{1 + \cos \left(a\right)} = \tan \left(a\right)$

#### Explanation:

$\frac{\sin \left(a\right) + \tan \left(a\right)}{1 + \cos \left(a\right)} = \frac{\left(\sin \left(a\right) + \tan \left(a\right)\right) \left(1 - \cos \left(a\right)\right)}{\left(1 + \cos \left(a\right)\right) \left(1 - \cos \left(a\right)\right)}$

$= \frac{\tan \left(a\right) - \sin \left(a\right) \cos \left(a\right)}{1 - {\cos}^{2} \left(a\right)}$

$= \frac{\tan \left(a\right) - \sin \left(a\right) \cos \left(a\right)}{\sin} ^ 2 \left(a\right)$

$= \frac{\cot \left(a\right) \left(\tan \left(a\right) - \sin \left(a\right) \cos \left(a\right)\right)}{\cot \left(a\right) {\sin}^{2} \left(a\right)}$

$= \frac{1 - {\cos}^{2} \left(a\right)}{\cos \left(a\right) \sin \left(a\right)}$

$= {\sin}^{2} \frac{a}{\cos \left(a\right) \sin \left(a\right)}$

$= \sin \frac{a}{\cos} \left(a\right)$

$= \tan \left(a\right)$