Class 12 Inverse Trigonometric Functions · Long Answer3 Marksinverse_trigonometric_identities
Class 12 Maths · Inverse Trigonometric Functions · Long Answer #70
Prove \(\cos\{\tan^{-1}[\sin(\cot^{-1}x)]\}=\sqrt{(1+x^2)/(2+x^2)}\), x real.
✅ Step-by-Step Solution
Final Answer: Identity proved for every real x.
- Let θ=arccot x∈(0,π), so sin θ>0 and sin θ=1/√(1+x²).
- Put u=sin θ=1/√(1+x²), and α=arctan u∈(0,π/2). Then cos α=1/√(1+u²).
- Substitute u²=1/(1+x²): cos α=1/√((2+x²)/(1+x²))=√((1+x²)/(2+x²)).
- This is the required positive square root, proving the identity on R.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let θ=arccot x∈(0,π), so sin θ>0 and sin θ=1/√(1+x²).
- [1 mark] Put u=sin θ=1/√(1+x²), and α=arctan u∈(0,π/2). Then cos α=1/√(1+u²).
- [1 mark] Substitute u²=1/(1+x²): cos α=1/√((2+x²)/(1+x²))=√((1+x²)/(2+x²)).
- [0.5 marks] This is the required positive square root, proving the identity on R.
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