Class 12 Inverse Trigonometric Functions · Long Answer3 Marksinverse_trigonometric_equations
Class 12 Maths · Inverse Trigonometric Functions · Long Answer #68
Solve \(\cos(\tan^{-1}x)=\sin[\cot^{-1}(3/4)]\) over R.
✅ Step-by-Step Solution
Final Answer: x=−3/4 or 3/4.
- For θ=arccot(3/4), θ is acute and sin θ=4/5.
- The principal arctangent lies in (−π/2,π/2), so cos(arctan x)=1/√(1+x²). Set this equal to 4/5.
- Both sides are positive; squaring gives 1+x²=25/16, so x²=9/16. Both x=−3/4 and 3/4 give cosine 4/5.
- The complete real solution set is {−3/4,3/4}.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] For θ=arccot(3/4), θ is acute and sin θ=4/5.
- [1 mark] The principal arctangent lies in (−π/2,π/2), so cos(arctan x)=1/√(1+x²). Set this equal to 4/5.
- [1 mark] Both sides are positive; squaring gives 1+x²=25/16, so x²=9/16. Both x=−3/4 and 3/4 give cosine 4/5.
- [0.5 marks] The complete real solution set is {−3/4,3/4}.
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