# Question 10fa9

Nov 14, 2017

$\frac{\cos \left(x + y\right) + \cos \left(x - y\right)}{\cos x \sin y} = 2 \cot y$

Apply the identity $\cos \left(\alpha \pm \beta\right) = \cos \alpha \cos \beta \pm \left(- \sin \alpha \sin \beta\right)$

$\frac{\cos x \cos y - \sin x \sin y + \cos x \cos y + \sin x \sin y}{\cos x \sin y}$
$\frac{2 \cos x \cos y}{\cos x \sin y}$
$\frac{2 \cos y}{\sin} y$

Definitionally, this is $2 \cot y$

Nov 14, 2017

$\text{see explanation}$

#### Explanation:

$\text{using the "color(blue)"addition formulae for cos}$

•color(white)(x)cos(A+-B)=cosAcosB∓sinAsinB

•color(white)(x)coty=cosy/siny#

$\text{consider the LHS}$

$\frac{\cos \left(x + y\right) + \cos \left(x - y\right)}{\cos x \sin y}$

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

$= \frac{2 \cos x \cos y}{\cos x \sin y}$

$= \frac{2 \cancel{\cos x} \cos y}{\cancel{\cos x} \sin y}$

$= 2 \cos \frac{y}{\sin} y$

$= 2 \cot y = \text{RHS"rArr" proved}$