Question

Establish each identity.

$\frac{\tan \alpha +\tan \beta }{cot\alpha +cot\beta }=\tan \alpha \tan \beta$

Short answer

The left side of the given equation has been simplified to bring the right side.

Step-by-step solution

Step 1. Given Information

The given equation is$\frac{\tan \alpha +\tan \beta }{cot\alpha +cot\beta }=\tan \alpha \tan \beta$

Step 2. Calculation 

Consider the left side of the given equation and simplify to bring the right side in order to establish the equation.

$$\begin{gathered} \frac{\tan \alpha +\tan \beta }{cot\alpha +cot\beta }=\frac{\tan \alpha +\tan \beta }{{\displaystyle \frac{1}{\tan \alpha }}+{\displaystyle \frac{1}{\tan \beta }}} \\ =\frac{\tan \alpha +\tan \beta }{{\displaystyle \frac{\tan \beta +\tan \alpha }{\tan \alpha \tan \beta }}} \\ =\tan \alpha \tan \beta \frac{(\tan \alpha +\tan \beta )}{(\tan \alpha +\tan \beta )} \\ =\tan \alpha \tan \beta \end{gathered}$$

Hence, Proved