# How do you prove (1+tany)/(1+coty)=secy/cscy?

Aug 25, 2016

See below.

#### Explanation:

Apply the following identities:

$\tan \theta = \sin \frac{\theta}{\cos} \theta$

$\cot \theta = \frac{1}{\tan} \theta = \frac{1}{\sin \frac{\theta}{\cos} \theta} = \cos \frac{\theta}{\sin} \theta$

$\sec \theta = \frac{1}{\cos} \theta$

$\csc \theta = \frac{1}{\sin} \theta$

Start the simplification process on both sides.

$\frac{1 + \sin \frac{y}{\cos} y}{1 + \cos \frac{y}{\sin} y} = \frac{\frac{1}{\cos} y}{\frac{1}{\sin} y}$

Put on a common denominator:

$\frac{\frac{\cos y + \sin y}{\cos} y}{\frac{\sin y + \cos y}{\sin} y} = \frac{1}{\cos} y \times \sin \frac{y}{1}$

$\frac{\cos y + \sin y}{\cos} y \times \sin \frac{y}{\sin y + \cos y} = \sin \frac{y}{\cos} y$

$\frac{\cancel{\cos y + \sin y}}{\cos} y \times \sin \frac{y}{\cancel{\sin y + \cos y}} = \sin \frac{y}{\cos} y$

$\sin \frac{y}{\cos} y = \sin \frac{y}{\cos} y$

Identity proved!!

Hopefully this helps!