What is the maximum value of this trigonometric expression What is the maximum value of the expression
$1/(\sin^2 \theta + 3\sin\theta \cos\theta+ 5\cos^2 \theta$).
I tried reducing the expression to $1/(1 + 3\sin\theta$ $\cos\theta + 4\cos^2 \theta)$. 
How do I proceed from here?

This is the same as minimizing the function $f(\theta)=\sin^2 \theta+3 \sin \theta \cos \theta+5\cos^2 \theta$ (subject to $f(\theta)>0$).  We know such a minimum will occur only when $f'(\theta)=0$.  We have
\begin{align*}
f'(\theta)&= 2 \sin \theta \cos \theta - 3 \sin^2 \theta +3 \cos^2 \theta-10 \sin \theta \cos \theta=-8 \sin \theta \cos \theta-3\sin^2 \theta + 3 \cos^2 \theta.
\end{align*}
Using trig identities, this is same as solving $3\cos 2\theta -4 \sin 2 \theta=0.$  The critical points occurs when $\tan 2 \theta= \frac{3}{4}$.  At such a point $\sin 2 \theta=\frac{\pm 3}{5}$ and $\cos 2 \theta=\frac{\pm 4}{ 5}.$  As a result, $\sin \theta \cos \theta= \frac{\pm 3}{ 10}$ and $\cos^2 \theta -\sin^2 \theta= \frac{\pm 4}{5}$.
Now we have $f(\theta)=2\cos^2 \theta - 2\sin^2 \theta+3 + 3 \sin \theta \cos \theta.$  Substituting the values of $\cos \theta \sin \theta$ and $\cos^2 \theta- \sin^2 \theta$ at the critical points, we can find max/min values for $f$. 
