NCERT Class 12 · Ch 2 Miscellaneous Exercise · Q9
NCERT Class 12 Maths · Chapter 2 Inverse Trigonometric Functions · Miscellaneous Exercise Question 9
Prove cot⁻¹((√(1+sin x)+√(1−sin x))/(√(1+sin x)−√(1−sin x)))=x/2 for 0<x<π/4.
💡 Concept Hints
- Recall the principal range of the inverse function before simplifying.
- Find an angle with the required trigonometric value, then verify it belongs to the principal range.
✅ Step-by-Step Solution
Final Answer: The identity holds.
- Write c=cos(x/2), s=sin(x/2). On this interval c>s>0.
- Since 1±sin x=(c±s)² and c±s>0, the square roots are c+s and c−s.
- The quotient becomes 2c/(2s)=cot(x/2). Since x/2∈(0,π), principal cot⁻¹ returns x/2.
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