Question

Establish each identity.

$(\tan \alpha +\tan \beta )(1-cot\alpha cot\beta )+(cot\alpha +cot\beta )(1-\tan \alpha \tan \beta )=0$

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$(\tan \alpha +\tan \beta )(1-cot\alpha cot\beta )+(cot\alpha +cot\beta )(1-\tan \alpha \tan \beta )=0$

Step 2. Calculation 

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

$$\begin{gathered} (\tan \alpha +\tan \beta )(1-cot\alpha cot\beta )+(cot\alpha +cot\beta )(1-\tan \alpha \tan \beta )=\tan \alpha -\tan \alpha cot\alpha cot\beta +\tan \beta -\tan \beta cot\alpha cot\beta +cot\alpha -cot\alpha \tan \alpha \tan \beta +cot\beta -cot\beta \tan \alpha \tan \beta \\ =\tan \alpha -\left(\frac{1}{cot\alpha }cot\alpha cot\beta \right)+\tan \beta -\left(\frac{1}{cot\beta }cot\alpha cot\beta \right)+cot\alpha -\left(\frac{1}{\tan \alpha }\tan \alpha \tan \beta \right)+cot\beta -\left(\frac{1}{\tan \beta }\tan \alpha \tan \beta \right) \\ =\tan \alpha -cot\beta +\tan \beta -cot\alpha +cot\alpha -\tan \beta +cot\beta -\tan \alpha \\ =0 \end{gathered}$$

Hence, Proved.