Question

Establish the identity:$1+{\tan }^2(-\theta )=se{c}^2\left(\theta \right)$

Short answer

We establish the identity$1+{\tan }^2(-\theta )=se{c}^2\left(\theta \right)$

Step-by-step solution

Given information

We are given$1+{\tan }^2(-\theta )$

tan is an odd function tan(-θ)=-tan(θ)

We get

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

Conclusion

We establish the identity$1+{\tan }^2(-\theta )=se{c}^2\left(\theta \right)$