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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?

wrb98
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    you can take a look at AOPS (art of problemsolving) – Dr. Sonnhard Graubner Jul 06 '17 at 09:22
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    A neat way to maximize $f(x) = \alpha x(1-x)$ (logistic mapping) on the interval $[0,1]$ without calculus is setting $x = \sin^2 \theta$ so we get $\alpha\sin^2\theta\cos^2\theta = \frac{\alpha}{4} \sin^2 2\theta$ – MathematicsStudent1122 Jul 06 '17 at 09:30
  • See also: https://math.stackexchange.com/questions/429883/roots-of-8x3-4x2-4x1 and https://math.stackexchange.com/questions/550052/proving-the-second-root-of-a-quadratic-equation/2258008#2258008 https://math.stackexchange.com/questions/2203364/solve-the-following-equation-x3-3x-sqrtx2 https://math.stackexchange.com/questions/2157643/how-can-i-solve-the-equation-x3-x-1-0/2157645#2157645 – lab bhattacharjee Jul 06 '17 at 09:42

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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

Jaideep Khare
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  • This may look artificial until you know the derivation : https://math.stackexchange.com/questions/2304904/how-to-integrate-int-ab-t-cdot-sqrt-fract-ab-t-dt/2305226#2305226 – lab bhattacharjee Jul 06 '17 at 09:40
  • @labbhattacharjee I don't understand the meaning of artificial. Can you please elaborate? – Jaideep Khare Jul 06 '17 at 09:43
  • How on earth, you know it and how much it is susceptible to generalization? – lab bhattacharjee Jul 06 '17 at 09:44
  • @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'. – Jaideep Khare Jul 06 '17 at 09:48
  • 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? https://math.stackexchange.com/questions/328562/divisibility-criteria-for-7-11-13-17-19 – lab bhattacharjee Jul 06 '17 at 09:52
  • @labbhattacharjee I agree to this point. That is leading to deteriorated condition of education in our country. BTW I knew that the proof of test of divisibility. That too wasn't told in the school, I read it on my own when I was preparing for KVPY Stage II (Interview round). Let's hope, one day, education system will change. – Jaideep Khare Jul 06 '17 at 09:59
  • I like this example, although I was looking for problems more similar to the one I posed. It'd be great if someone could come up with some creative examples of an equation that could be simplified with this substitution. – wrb98 Jul 06 '17 at 10:24