# How do you prove cos(u-v)/(cosusinv)=tanu+cotv?

Sep 4, 2016

see explanation

#### Explanation:

Attempt to convert the left side into the form of the right side.

Consider the numerator of the function on the left. Using the appropriate $\textcolor{b l u e}{\text{addition formula}}$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\cos \left(A - B\right) = \cos A \cos B + \sin A \sin B} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\Rightarrow \cos \left(u - v\right) = \cos u \cos v + \sin u \sin v$

We now have : $\frac{\cos u \cos v + \sin u \sin v}{\cos u \sin v}$

now divide the terms on the numerator by the denominator.

$\Rightarrow \frac{\cancel{\cos u} \cos v}{\cancel{\cos u} \sin v} + \frac{\sin u \cancel{\sin v}}{\cos u \cancel{\sin v}} = \frac{\cos v}{\sin v} + \frac{\sin u}{\cos u}$

color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(tantheta=(sintheta)/(costheta)" and " cottheta=(costheta)/(sintheta))color(white)(a/a)|)))

$\Rightarrow \frac{\cos v}{\sin v} + \frac{\sin u}{\cos u} = \cot v + \tan u = \tan u + \cot v$

Thus left side = right side $\Rightarrow \text{ proven}$