NCERT Class 12 · Ch 2 Miscellaneous Exercise · Q8
NCERT Class 12 Maths · Chapter 2 Inverse Trigonometric Functions · Miscellaneous Exercise Question 8
Prove tan⁻¹√x=(1/2)cos⁻¹((1−x)/(1+x)), 0≤x≤1.
💡 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.
- Set θ=tan⁻¹√x∈[0,π/4].
- cos 2θ=(1−tan²θ)/(1+tan²θ)=(1−x)/(1+x).
- Since 2θ∈[0,π/2]⊂[0,π], cos⁻¹((1−x)/(1+x))=2θ. Divide by 2.
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