An inverse-trigonometric identity and its endpoints

(i) Show that ddx(cos⁡−12x1+x2)=−21+x2\frac{d}{dx}\left(\cos^{-1}\frac{2x}{1+x^2}\right)=\frac{-2}{1+x^2} for −1≤x≤1-1\le x\le1. Explain, in your working, how you use the condition −1≤x≤1-1\le x\le1 to conclude your answer. marks

(ii) Hence, determine the constants AA and BB satisfying the equation cos⁡−12x1+x2+Atan⁡−1x=B\cos^{-1}\frac{2x}{1+x^2}+A\tan^{-1}x=B, for −1≤x≤1-1\le x\le1. marks

[2009/YJC/Prelim/I/5(a)]

Total marks: 6

Solution

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  1. (i) Chain rule with inverse trigonometry →
  2. (ii) Constants from derivatives →