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Class 12 Inverse Trigonometric Functions · Long Answer3 Marksinverse_trigonometric_identities

Class 12 Maths · Inverse Trigonometric Functions · Long Answer #74

For \(0<x<\pi/4\), prove \(\cot^{-1}\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}=x/2\).

✅ Step-by-Step Solution

Final Answer: Identity proved on (0,π/4).
  1. For 0<x/2<π/8, cos(x/2)±sin(x/2) are both positive.
  2. Thus √(1+sin x)=cos(x/2)+sin(x/2) and √(1−sin x)=cos(x/2)−sin(x/2).
  3. The quotient of their sum and difference is [2cos(x/2)]/[2sin(x/2)]=cot(x/2). Since x/2∈(0,π), it is on the arccot principal branch.
  4. Therefore the left side equals x/2.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] For 0<x/2<π/8, cos(x/2)±sin(x/2) are both positive.
  • [1 mark] Thus √(1+sin x)=cos(x/2)+sin(x/2) and √(1−sin x)=cos(x/2)−sin(x/2).
  • [1 mark] The quotient of their sum and difference is [2cos(x/2)]/[2sin(x/2)]=cot(x/2). Since x/2∈(0,π), it is on the arccot principal branch.
  • [0.5 marks] Therefore the left side equals x/2.

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