Trigonometry & Inverse Trigonometry
Solution Set of arcsin Difference Equation
nta_pyq_2023_apr
Grade 12
Question:
If $S=\left\{x\in\mathbb{R}:\sin^{-1}\!\left(\dfrac{x+1}{\sqrt{x^2+2x+2}}\right)-\sin^{-1}\!\left(\dfrac{x}{\sqrt{x^2+1}}\right)=\dfrac{\pi}{4}\right\}$, then $\displaystyle\sum_{x\in S}\left(\sin\frac{(x^2+x+5)\pi}{2}-\cos\frac{(x^2+x+5)\pi}{2}\right)$ is equal to _________.
Step-by-Step Solution
Key Concept: Use the formula $\sin^{-1}A-\sin^{-1}B=\frac{\pi}{4}$ and the identity for difference of arcsines. After simplification get $(x+1)(\sqrt{2}\sqrt{x^2+1}-\sqrt{x^2+x+2})=0$.
$S=\{-1,0\}$. Each term $=2$. Sum $=4$.
Correct Answer: 4