IIT-JEE 2005 ScreeningQuestion 1
$\cos(\alpha-\beta)=1$ and $\cos(\alpha+\beta)=1/e$ where $\alpha,\beta\in[-\pi,\pi]$. The number of pairs $\alpha,\beta$ which satisfy both the equations is
Trigonometric Ratio & Identities
Solution
Concept. $\cos(\alpha-\beta)=1$ forces $\alpha=\beta$; then count the solutions of $\cos2\alpha=\frac1e$ on a graph. Formulas used. (1) $\cos\varphi=1\iff\varphi=2k\pi$Step 1 - $\alpha=\beta$. $\alpha-\beta\in[-2\pi,2\pi]$, so by (1) $\alpha-\beta=0$ or $\pm2\pi$. The case $\pm2\pi$ means $\{\alpha,\beta\}=\{\pi,-\pi\}$, so $\alpha+\beta=0$ and $\cos(\alpha+\beta)=1\ne\frac1e$. Hence $\alpha=\beta$. Step 2 - The second equation. $\cos2\alpha=\frac1e$ with $u=2\alpha\in[-2\pi,2\pi]$. Since $0<\frac1e<1$, the line $y=\frac1e$ cuts $y=\cos u$ four times:Step 3 - Count. Each $u$ gives one pair $(\alpha,\beta)=\left(\frac u2,\frac u2\right)$: $4$ pairs. Answer: (D)