Trigonometry & Inverse Trigonometry
Finding k Then Counting Solutions of sin⁻¹(kx−1)
nta_pyq_2026_jan
Grade 12
Question:
If $k=\tan\!\left(\dfrac{\pi}{4}+\dfrac{1}{2}\cos^{-1}\!\dfrac{2}{3}\right)+\tan\!\left(\dfrac{1}{2}\sin^{-1}\!\dfrac{2}{3}\right)$, then the number of solutions of the equation $\sin^{-1}(kx-1)=\sin^{-1}x-\cos^{-1}x$ is _____.
Step-by-Step Solution
Key Concept: Let $\theta=\tfrac{1}{2}\sin^{-1}\tfrac{2}{3}$. Then $k=\tan\theta+\cot\theta=\tfrac{1}{\sin\theta\cos\theta}=\tfrac{2}{\sin2\theta}=\tfrac{2}{2/3}=3$. Equation: $\sin^{-1}(3x-1)=\sin^{-1}x-\cos^{-1}x=\tfrac{\pi}{2}-2\cos^{-1}x$.
$k=3$. Number of solutions $=1$.
Correct Answer: 1