Trigonometrical Ratios and Identities 1 Question 9
9. The number of ordered pairs $(\alpha, \beta)$, where $\alpha, \beta \in(-\pi, \pi)$ satisfying $\cos (\alpha-\beta)=1$ and $\cos (\alpha+\beta)=\frac{1}{e}$ is
(2005, 1M)
(a) 0
(b) 1
(c) 2
(d) 4
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Answer:
Correct Answer: 9. (b)
Solution:
- Since, $\quad \cos (\alpha-\beta)=1$
$\begin{array}{lrr}\Rightarrow & \alpha-\beta=2 n \pi & \ \text { But } & -2 \pi<\alpha-\beta<2 \pi & {[\text { as } \alpha, \beta \in(-\pi, \pi)]} \ \therefore & \alpha-\beta=0 & \text {…(i) } \ \text { Given, } & \cos (\alpha+\beta)=\frac{1}{e}\end{array}$
$\Rightarrow \cos 2 \alpha=\frac{1}{e}<1$, which is true for four values of $\alpha$.
[as $-2 \pi<2 \alpha<2 \pi]$
