TITLE: Examples of trigonometric substitutions for solving equations
QUESTION [5 upvotes]: I was working through a booklet of Olympiad-style problems when I came across a method which used the substitution $x = \cos \alpha$ to solve $x = \sqrt{2 + \sqrt{2-\sqrt{2+x}}}$. The solution works out nicely using the half angle formula. Are there any other good examples of such equations, where a trigonometric substitution and an identity can reduce a problem like this so effectively?

REPLY [2 votes]: The substitution 
$$t=a\cos^2 \theta+b \sin^2 \theta$$
Simplifies the function $\displaystyle \sqrt{\frac{t - a}{b - t}}$ tremendously to $\tan \theta$. 
You can see application of this substitution here