What are some useful tricks/shortcuts for verifying trigonometric identities? What "tricks" are there that could help verify trigonometric identities? 
For example one is:
$$a\cos\theta+b\sin\theta = \sqrt{a^2+b^2}\,\cos(\theta-\phi)$$        

Expand the cosine of the difference of angles on the right.
$$ \cos(\theta - \phi) = \cos \theta \cos \phi + \sin \theta \sin \phi$$
Now, collect terms so that the expression is a linear combination of $\sin \theta$ and $\cos \theta$, as in the expression on the left.
$$ \begin{align}
\sqrt{a^2 + b^2}\cos(\theta - \phi) &= \sqrt{a^2 + b^2}(\cos \theta \cos \phi + \sin \theta \sin \phi)\\
&=\sqrt{a^2 + b^2}\cos \phi \cdot \cos \theta + \sqrt{a^2 + b^2}\sin \phi \cdot \sin \theta\\
&=a \cos \theta + b \sin \theta
\end{align}$$
This last equality holds as long as we choose $\phi$ such that
$$ \left\{\begin{align} \cos \phi &= \frac{a}{\sqrt{a^2 + b^2}}\\ \sin \phi &=  \frac{b}{\sqrt{a^2 + b^2}} \end{align}\right.$$
This is always possible since those are the coordinates of a point on the unit circle.
