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

Note that $\cos(x-y)=\cos x\cos y+\sin x\sin y$.  This is obtained from the more familiar formula for $\cos(x+y)$ by replacing $y$ by $-y$. 
Note also that 
$$a\cos\theta+b\sin\theta=\sqrt{a^2+b^2}\left(\frac{a}{\sqrt{a^2+b^2}}\cos\theta+   \frac{b}{\sqrt{a^2+b^2}}\sin\theta   \right).$$
So if $\phi$ is the angle whose cosine is $\frac{a}{\sqrt{a^2+b^2}}$ and whose sine is $\frac{b}{\sqrt{a^2+b^2}}$, then in the formula above we can replace  $\frac{a}{\sqrt{a^2+b^2}}$ by $\cos\phi$, and  $\frac{b}{\sqrt{a^2+b^2}}$ with $\sin\phi$, and obtain
$$a\cos\theta+b\sin\theta=\sqrt{a^2+b^2} \cos(\theta-\phi).$$
Remark: As to tricks and shortcuts, mostly it is a question of experience and practice. Already, I am sure, you recognize certain patterns and know how to exploit them. After a while, you will have used most of the common devices a dozen times, and then things get easy.  
