Make the Trigonometric Substitution $$x = a \tan \theta \text { for } \frac { - \pi } { 2 } < \theta < \frac { \pi } { 2 }$$

Multiple Choice

Make the trigonometric substitution $$x = a \tan \theta \text { for } \frac { - \pi } { 2 } < \theta < \frac { \pi } { 2 }$$ and a > 0. Use fundamental identities to simplify the resulting expression. $\frac { 1 } { x ^ { 2 } + a ^ { 2 } }$

A) $a ^ { 2 } \sec ^ { 2 } \theta \csc \theta$
B) $\frac { 1 } { a } \cos ^ { 2 } \theta$
C) $\frac { 1 } { a } \cos \theta$
D) $\frac { 1 } { a ^ { 2 } } \cos ^ { 2 } \theta$
E) $a \sec \theta$