Class 12 Inverse Trigonometric Functions · Long Answer3 Marksinverse_trigonometric_identities
Class 12 Maths · Inverse Trigonometric Functions · Long Answer #69
Prove \(\tan[\tfrac12\sin^{-1}(3/4)]=(4-\sqrt7)/3\).
✅ Step-by-Step Solution
Final Answer: Identity proved.
- Let θ=arcsin(3/4)∈(0,π/2), so cos θ=√7/4.
- The half-angle identity gives tan(θ/2)=sin θ/(1+cos θ).
- Substituting gives tan(θ/2)=(3/4)/(1+√7/4)=3/(4+√7)=(4−√7)/3.
- Thus tan(½arcsin(3/4))=(4−√7)/3, with the positive acute half-angle branch justified.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let θ=arcsin(3/4)∈(0,π/2), so cos θ=√7/4.
- [1 mark] The half-angle identity gives tan(θ/2)=sin θ/(1+cos θ).
- [1 mark] Substituting gives tan(θ/2)=(3/4)/(1+√7/4)=3/(4+√7)=(4−√7)/3.
- [0.5 marks] Thus tan(½arcsin(3/4))=(4−√7)/3, with the positive acute half-angle branch justified.
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