Class 12 Inverse Trigonometric Functions · Short Answer2 Marksbranch_sensitive_simplification
Class 12 Maths · Inverse Trigonometric Functions · Short Answer #91
For a>0 and |x|<a, simplify \(\tan^{-1}[x/\sqrt{a^2-x^2}]\).
✅ Step-by-Step Solution
Final Answer: sin⁻¹(x/a)
- Let θ=arcsin(x/a). Since |x|<a and a>0, θ∈(−π/2,π/2).
- Its cosine is √(a²−x²)/a>0; hence tan θ=(x/a)/(√(a²−x²)/a)=x/√(a²−x²).
- Taking the principal arctangent gives θ=arcsin(x/a).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let θ=arcsin(x/a). Since |x|<a and a>0, θ∈(−π/2,π/2).
- [1 mark] Its cosine is √(a²−x²)/a>0; hence tan θ=(x/a)/(√(a²−x²)/a)=x/√(a²−x²).
- [0.5 marks] Taking the principal arctangent gives θ=arcsin(x/a).
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