Answer
I know that $\tan$ is an odd function, which means $f(-\theta)=-f(\theta).$ Therefore $\tan{-\theta}=-\tan{\theta}$.
Thus: $1+\tan^2{(-\theta)}=1+(-\tan{(-\theta)})^2=1+\tan^2{\theta}.$
We know that $1+\tan^2{\theta}=\sec^2{\theta}$, hence we proved the identity.
Work Step by Step
I know that $\tan$ is an odd function, which means $f(-\theta)=-f(\theta).$ Therefore $\tan{-\theta}=-\tan{\theta}$.
Thus: $1+\tan^2{(-\theta)}=1+(-\tan{(-\theta)})^2=1+\tan^2{\theta}.$
We know that $1+\tan^2{\theta}=\sec^2{\theta}$, hence we proved the identity.
