Class 12 Inverse Trigonometric Functions · Long Answer3 Marksbranch_sensitive_simplification
Class 12 Maths · Inverse Trigonometric Functions · Long Answer #103
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 −1<x/a<1, θ lies in (−π/2,π/2).
- The principal cosine is cos θ=√(1−x²/a²)=√(a²−x²)/a, using a>0.
- Therefore tan θ=(x/a)/(√(a²−x²)/a)=x/√(a²−x²). Because θ is principal, arctan(tan θ)=θ.
- The simplified expression is arcsin(x/a).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let θ=arcsin(x/a). Since −1<x/a<1, θ lies in (−π/2,π/2).
- [1 mark] The principal cosine is cos θ=√(1−x²/a²)=√(a²−x²)/a, using a>0.
- [1 mark] Therefore tan θ=(x/a)/(√(a²−x²)/a)=x/√(a²−x²). Because θ is principal, arctan(tan θ)=θ.
- [0.5 marks] The simplified expression is arcsin(x/a).
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