Question

True or False

$\tan \left(2\theta \right)=\frac{2\tan \theta }{1-{\tan }^2\theta }$

Short answer

Equation$\tan \left(2\theta \right)=\frac{2\tan \theta }{1-{\tan }^2\theta }$is true

Step-by-step solution

Step 1. Given Data   

The given equation is

$\tan \left(2\theta \right)=\frac{2\tan \theta }{1-{\tan }^2\theta }$

Step 2. Explanation

Use sum formula for $\tan (\alpha +\beta )$

role="math" localid="1646429460524" $\tan \left(\alpha +\beta \right)=\frac{\tan \alpha +\tan \beta }{1-\left(\tan \alpha \right)\left(\tan \beta \right)}$

Substitute $\theta$forrole="math" localid="1646429474469" $\beta \&\alpha$

$$\begin{gathered} \tan \left(\theta +\theta \right)=\frac{\tan \theta +\tan \theta }{1-\left(\tan \theta \right)\left(\tan \theta \right)} \\ \tan \left(2\theta \right)=\frac{2\tan \theta }{1-{\tan }^2\theta } \end{gathered}$$

so the given equation is true