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?

Another method is to rewrite $\sin^2 \theta + 3 \sin \theta \cos \theta + 5 \cos^2\theta$ as
$$\frac{1 + (\sin\theta + 3\cos\theta)^2}{2}.$$
The minimum value of this expression is $\frac12$, so the maximum value of the original expression is 2.
