I feel like my approach thus far is valid, but I've reached a spot where it feel like things are going wrong. I've changed the $1/\sin^2 \to \csc^2$ and distributed it to make it easier to integrate. $\sin^2 \times \csc^2 = 1$, so those cancel... but that leaves me with a constant $6$, which if my variable were $x$ I'd feel right at home. But this is $\theta$, and making that $6$ into a $6 \theta$ just seems like I've gone wrong somewhere.
Can anyone give me a push in the right direction?
Here's my approach so far:
$$ \eqalign { \text {Question: } & \int \frac { 7 - 6 \sin ^ 2 ( \theta ) } { \sin ^ 2 ( \theta ) } \, d \theta \\ \text {My answer: } & \int ( 7 - 6 \sin ^ 2 \theta ) ( \csc ^ 2 \theta ) \, d \theta \\ & \int ( 7 \csc ^ 2 \theta - 6 \sin ^ 2 \theta \csc ^ 2 \theta ) \, d \theta \\ & \int ( 7 \csc ^ 2 \theta - 6 ) \, d \theta \\ & 7 \cot \theta \> \cdots \hfil } $$