Class 12 Inverse Trigonometric Functions · Short Answer2 Marksbranch_sensitive_simplification
Class 12 Maths · Inverse Trigonometric Functions · Short Answer #59
For |x|>1, simplify \(\cot^{-1}[1/\sqrt{x^2-1}]\).
✅ Step-by-Step Solution
Final Answer: arcsec|x|; equivalently arcsec x for x>1 and π−arcsec x for x<−1.
- Let u=|x|>1 and θ=arcsec u∈(0,π/2). Then tan θ=√(u²−1)=√(x²−1).
- Hence 1/√(x²−1)=cot θ. Since θ is in the arccot principal range (0,π), arccot(cot θ)=θ.
- The result is arcsec|x|. It is arcsec x for x>1 and π−arcsec x for x<−1.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let u=|x|>1 and θ=arcsec u∈(0,π/2). Then tan θ=√(u²−1)=√(x²−1).
- [1 mark] Hence 1/√(x²−1)=cot θ. Since θ is in the arccot principal range (0,π), arccot(cot θ)=θ.
- [0.5 marks] The result is arcsec|x|. It is arcsec x for x>1 and π−arcsec x for x<−1.
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