# Question #acc87

Nov 6, 2017

Please refer to a Proof in the Explanation.

#### Explanation:

$\frac{\tan x + \sin x}{1 + \cos x} ,$

$= \frac{\sin \frac{x}{\cos} x + \sin x}{1 + \cos x} ,$

$= \frac{\frac{\sin x + \sin x \cos x}{\cos} x}{1 + \cos x} ,$

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

$= \sin \frac{x}{\cos} x ,$

$= \tan x .$

Hence, the Proof.

Nov 6, 2017

See the demonstration below:

#### Explanation:

Start with the equation: $\frac{\tan \left(x\right) + \sin \left(x\right)}{1 + \cos \left(x\right)} = \tan \left(x\right)$

Move $1 + \cos \left(x\right)$ on the other side of the equal sign: $\tan \left(x\right) + \sin \left(x\right) = \left(1 + \cos \left(x\right)\right) \cdot \tan \left(x\right)$

Expand $\left(1 + \cos \left(x\right)\right) \cdot \tan \left(x\right) = \tan \left(x\right) + \tan \left(x\right) \cdot \cos \left(x\right)$

Use the equality $\tan \left(x\right) = \sin \frac{x}{\cos} \left(x\right)$: $\tan \left(x\right) + \sin \left(x\right) = \tan \left(x\right) + \sin \frac{x}{\cos} \left(x\right) \cdot \cos \left(x\right)$

Simplify: $\tan \left(x\right) + \sin \left(x\right) = \tan \left(x\right) + \sin \frac{x}{\cancel{\cos \left(x\right)}} \cdot \cancel{\cos \left(x\right)}$

You are left with: $\tan \left(x\right) + \sin \left(x\right) = \tan \left(x\right) + \sin \left(x\right)$ which is true, proving that the starting equation is also true.