NCERT Class 12 · Ch 2 Miscellaneous Exercise · Q11
NCERT Class 12 Maths · Chapter 2 Inverse Trigonometric Functions · Miscellaneous Exercise Question 11
Solve 2tan⁻¹(cos x)=tan⁻¹(2cosec x) over all real x for which the equation is defined.
💡 Concept Hints
- State where cosec x exists, and locate both sides in principal-angle intervals.
- Taking tangent is reversible when both angles lie in the same interval where tangent is one-one.
✅ Step-by-Step Solution
Final Answer: x=π/4+kπ, k∈ℤ.
- The domain requires sin x≠0, hence −1<cos x<1 and 2tan⁻¹(cos x)∈(−π/2,π/2). The right side also lies in this range.
- Tangent is one-one there. Taking tangent gives 2cos x/(1−cos²x)=2/sin x.
- As sin x≠0, multiply by sin²x to obtain cos x=sin x, whose real solutions are x=π/4+kπ.
- For these values sin x≠0. Both original sides are in the same principal interval and have equal tangent, so every listed value satisfies the original equation.
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