\( \Large sin \theta =\frac{a^{2}-1}{a^{2}+1} \)

In, \( \Large \triangle ABC \)

\( \Large BC=\sqrt{AC^{2}-AB^{2}} \)

= \( \Large \sqrt{(a^{2}+1)^{2}-(a^{2}-1)^{2}} \)

= \( \Large \sqrt{a^{2}+1+2a^{2}-a^{2}-1+2a^{2}} \)

= \( \Large \sqrt{4a^{2}}=2a \)

\( \Large sec \theta +tan \theta =\frac{a^{2}+1}{2a}+\frac{a^{2}-1}{2a} \)

=\( \Large \frac{2a^{2}}{2a}=a \)