# How do you simplify (5+9cosx) / (sinx) + (9sinx) / (1+cosx)?

Jul 17, 2016

$\frac{5 + 9 \cos x}{\sin x} + \frac{9 \sin x}{1 + \cos x} = 14 \csc x$

#### Explanation:

We can start off this problem by getting a common denominator.

$\frac{5 + 9 \cos x}{\sin x} + \frac{9 \sin x}{1 + \cos x}$

$= \frac{\left(5 + 9 \cos x\right) \left(1 + \cos x\right) + 9 {\sin}^{2} x}{\sin x \left(1 + \cos x\right)}$

Multiplying everything out gives us

$= \frac{5 + 5 \cos x + 9 \cos x + 9 {\cos}^{2} x + 9 {\sin}^{2} x}{\sin x \left(1 + \cos x\right)}$

Factoring out a $9$ yields

$= \frac{5 + 14 \cos x + 9 \left({\cos}^{2} x + {\sin}^{2} x\right)}{\sin x \left(1 + \cos x\right)}$

$= \frac{5 + 14 \cos x + 9}{\sin x \left(1 + \cos x\right)}$

Adding like terms of $9$ and $5$, which equals to $14$ results in

$= \frac{14 + 14 \cos x}{\sin x \left(1 + \cos x\right)}$

Factoring out a $14$ yields

$= \frac{14 \cancel{\left(1 + \cos x\right)}}{\sin x \cancel{\left(1 + \cos x\right)}} = \frac{14}{\sin x} = 14 \csc x$

Jul 17, 2016

$14 \csc x$

#### Explanation:

(5+9cosx)/sinx+(9sinx)/((1+cosx)

$= \frac{5 + 9 \cos x}{\sin} x + \frac{9 \sin x \left(1 - \cos x\right)}{\left(1 + \cos x\right) \left(1 - \cos x\right)}$

$= \frac{5 + 9 \cos x}{\sin} x + \frac{9 \sin x \left(1 - \cos x\right)}{1 - {\cos}^{2} x}$

$= \frac{5 + 9 \cos x}{\sin} x + \frac{9 \sin x \left(1 - \cos x\right)}{\sin} ^ 2 x$

$= \frac{5 + 9 \cos x}{\sin} x + \frac{9 \left(1 - \cos x\right)}{\sin} x$

$= \left(\frac{5}{\sin} x + \frac{9 \cos x}{\sin} x\right) + \left(\frac{9}{\sin} x - \frac{9 \cos x}{\sin} x\right)$

$= 5 \csc x + 9 \cot x + 9 \csc x - 9 \cot x$

$= 14 \csc x$