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

If this part uses an earlier result, revisit (i) Chain rule with inverse trigonometry or open all working together.

(ii)

Constants from derivatives

Hence determine the constants AA and BB for which cos⁡−12x1+x2+Atan⁡−1x=B\cos^{-1}\dfrac{2x}{1+x^2}+A\tan^{-1}x=B for −1≤x≤1-1\le x\le1.

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Hint

  1. Differentiate both sides: the right-hand side is a constant, so its derivative is 0.

  2. Use part (i) and ddxtan⁡−1x=11+x2\frac d{dx}\tan^{-1}x=\frac1{1+x^2}.

  3. Find BB with a convenient xx.

Answer · (ii)

A=2,B=π/2A=2,B=\pi/2.

Identity holds throughout −1≤x≤1.

Explore the idea

Open interactive exploration

cos⁻¹(2x/(1 + x²)) + 2tan⁻¹x is flat only on [−1, 1]

Inside the interval the sum is the constant π/2. Outside, the absolute value changes sign and the sum grows.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • Sum on [−1, 1]
  • Y = π/2
Ideas to try
What is the sum at x=2x=2? Does it equal π2\frac\pi2?