# Prove trigonometric identities?

## Oh man, I am so lost! $\cos \frac{x - y}{\cos x \cos y} = 1 + \tan x \tan y$ Any advice on where to start?

Jun 10, 2018

See explanation

#### Explanation:

We want to prove

$\cos \frac{x - y}{\cos \left(x\right) \cos \left(y\right)} = 1 + \tan \left(x\right) \tan \left(y\right)$

Remember the angle-difference identity

color(blue)((1) color(white)(BB)cos(x-y)=cos(x)cos(y)+sin(x)sin(y)

Thus

$L H S = \cos \frac{x - y}{\cos \left(x\right) \cos \left(y\right)}$

$\textcolor{w h i t e}{L H S} = \frac{\cos \left(x\right) \cos \left(y\right) + \sin \left(x\right) \sin \left(y\right)}{\cos \left(x\right) \cos \left(y\right)} \leftarrow \text{(1)}$

$\textcolor{w h i t e}{L H S} = \frac{\cos \left(x\right) \cos \left(y\right)}{\cos \left(x\right) \cos \left(y\right)} + \frac{\sin \left(x\right) \sin \left(y\right)}{\cos \left(x\right) \cos \left(y\right)}$

$\textcolor{w h i t e}{L H S} = 1 + \left(\frac{\sin \left(x\right)}{\cos} \left(x\right)\right) \left(\sin \frac{y}{\cos} \left(y\right)\right)$

$\textcolor{w h i t e}{L H S} = 1 + \tan \left(x\right) \tan \left(y\right) = R H S$

Jun 10, 2018

$\text{see explanation}$

#### Explanation:

$\text{using the "color(blue)"trigonometric identity}$

•color(white)(x)cos(x-y)=cosxcosy+sinxsiny

$\text{consider the left side}$

$\frac{\cos x \cos y + \sin x \sin y}{\cos x \cos y}$

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

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

$= 1 + \tan x \tan y = \text{ right side"rArr"verified}$