NCERT Class 12 · Ch 2 Miscellaneous Exercise · Q10
NCERT Class 12 Maths · Chapter 2 Inverse Trigonometric Functions · Miscellaneous Exercise Question 10
Prove tan⁻¹((√(1+x)−√(1−x))/(√(1+x)+√(1−x)))=π/4−(1/2)cos⁻¹x for −1/√2≤x≤1.
💡 Concept Hints
- Recall the principal range of the inverse function before simplifying.
- Find an angle with the required trigonometric value, then verify it belongs to the principal range.
✅ Step-by-Step Solution
Final Answer: The identity holds.
- Put θ=(1/2)cos⁻¹x. Then 0≤θ≤3π/8, so sin θ≥0 and cos θ>0.
- √(1+x)=√2 cos θ and √(1−x)=√2 sin θ. The quotient is (cos θ−sin θ)/(cos θ+sin θ)=tan(π/4−θ).
- Here π/4−θ∈[−π/8,π/4], inside the arctangent range. Taking principal tan⁻¹ gives π/4−θ.
Master Chapter 2: Inverse Trigonometric Functions
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Inverse Trigonometric Functions.
← Previous Question
Prove cot⁻¹((√(1+sin x)+√(1−sin x))/(√(1+sin x)−√(1−sin x)))=x/2 for…
Next Question →Solve 2tan⁻¹(cos x)=tan⁻¹(2cosec x) over all real x for which the…
All Questions in Inverse Trigonometric Functions — Miscellaneous Exercise
Related Questions in Inverse Trigonometric Functions
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\sin^{-1}(\cos(13\pi/5))\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Find the full real solution set of \(\sin^{-1}x>\cos^{-1}x\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\sin[\pi/2-\sin^{-1}(-1/2)]\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
For \(x\in[-1,1]\setminus\{0\}\), simplify \(\cot(\sin^{-1}x)\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\sin^{-1}(\cos(\pi/9))\).
Class 12 Inverse Trigonometric Functions · Mcq Single Correct
Evaluate \(\theta=\sin^{-1}(\sin(-600^\circ))\), giving the answer in radians.