# Primitive of Reciprocal of Square of Sine of a x by Square of Cosine of a x

## Theorem

$\displaystyle \int \frac {\d x} {\sin^2 a x \cos^2 a x} = \frac {-2 \cot 2 a x} a + C$

## Proof

 $\displaystyle \int \frac {\d x} {\sin^2 a x \cos^2 a x}$ $=$ $\displaystyle \int \frac {\d x} {\left({\sin a x \cos a x}\right)^2}$ $\displaystyle$ $=$ $\displaystyle \int \frac {\d x} {\left({\frac {\sin 2 a x} 2}\right)^2}$ Double Angle Formula for Sine $\displaystyle$ $=$ $\displaystyle 4 \int \frac {\d x} {\sin^2 2 a x}$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle 4 \int \csc^2 2 a x \rd x$ Cosecant is Reciprocal of Sine $\displaystyle$ $=$ $\displaystyle 4 \frac {-\cot 2 a x} {2 a} + C$ Primitive of Square of Cosecant of a x $\displaystyle$ $=$ $\displaystyle \frac {-2 \cot 2 a x} a + C$ simplifying

$\blacksquare$