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Q.
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
IIT JEEIIT JEE 2005
Solution:
$Since, cos(\alpha-\beta)=1 $
$\Rightarrow \alpha-\beta=2n\pi$
But $ -2\pi < \alpha-\beta < 2\pi [as \alpha, \beta \in (-\pi, \pi)]$
$\therefore \alpha-\beta=0 ...(i)$
Given, $ co (\alpha+\beta)=\frac{1}{e}$
$\Rightarrow cos 2\alpha=\frac{1}{e} < 1, which is true for four values of \alpha.$
$ [as -2\pi < 2\alpha < 2\pi]$