Class 12 Inverse Trigonometric Functions · Long Answer3 Marksinverse_trigonometric_identities
Class 12 Maths · Inverse Trigonometric Functions · Long Answer #71
Prove \(\tan^{-1}[\cos x/(1+\sin x)]=\pi/4-x/2\) for \(-\pi/2<x<\pi/2\).
✅ Step-by-Step Solution
Final Answer: Identity proved on the stated interval.
- Let u=π/4−x/2. The domain gives 0<u<π/2.
- Half-angle formulas, or multiplying by (1−sin x), give cos x/(1+sin x)=tan(π/4−x/2)=tan u; the denominator is positive.
- Since u is in the principal arctangent interval, arctan(tan u)=u.
- Therefore the expression equals π/4−x/2.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let u=π/4−x/2. The domain gives 0<u<π/2.
- [1 mark] Half-angle formulas, or multiplying by (1−sin x), give cos x/(1+sin x)=tan(π/4−x/2)=tan u; the denominator is positive.
- [1 mark] Since u is in the principal arctangent interval, arctan(tan u)=u.
- [0.5 marks] Therefore the expression equals π/4−x/2.
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