Solution
= Solution
The <tangent half-angle substitution> gives $\sin\theta=2t/(1+t^2)$ and $\cos\theta=(1-t^2)/(1+t^2)$. Dividing and then taking the reciprocal square of the <cosine> gives
$$
\boxed{\tan\theta=\frac{2t}{1-t^2},\qquad\sec^2\theta=\frac{(1+t^2)^2}{(1-t^2)^2}.}
$$
These formulas apply where $t$ is defined and $t\ne\pm1$; at $t=\pm1$ the original <tangent> and <secant function> are undefined because $\cos\theta=0$.