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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?

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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

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  • $\begingroup$ This may look artificial until you know the derivation : math.stackexchange.com/questions/2304904/… $\endgroup$ Jul 6, 2017 at 9:40
  • $\begingroup$ @labbhattacharjee I don't understand the meaning of artificial. Can you please elaborate? $\endgroup$ Jul 6, 2017 at 9:43
  • $\begingroup$ How on earth, you know it and how much it is susceptible to generalization? $\endgroup$ Jul 6, 2017 at 9:44
  • $\begingroup$ @labbhattacharjee Actually, it was taught to us in "Indefinite Integration $\rightarrow$ Integration by substitution". I have memorized it as it is and never knew about it's 'derivation'. $\endgroup$ Jul 6, 2017 at 9:48
  • $\begingroup$ Exactly, therein lies the problem! "How/why" is missing in many places in our curriculum. For example, many know how to check for divisibility by $3$ but how many know why the rule works? math.stackexchange.com/questions/328562/… $\endgroup$ Jul 6, 2017 at 9:52

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