Question

$cos(\alpha -\beta )=1$ and $cos(\alpha +\beta )=\frac{1}{e}$, where $\alpha ,\beta \in [-\pi ,\pi ]$. Number of pairs of $\alpha ,\beta$ which satisfy both the equation is

• A. \N
• B. 1
• C. 2
• D. 4

Answer 1

The correct option is D 4

Given that $cos(\alpha -\beta )=1$ and cos $(\alpha +\beta )=\frac{1}{e}$
Where $\alpha ,\beta \in [-\pi ,\pi ]$
Now $cos(\alpha -\beta )=1\Rightarrow \alpha -\beta =0$ or $\alpha =\beta$
$\therefore cos(\alpha +\beta )=\frac{1}{e}\Rightarrow cos2\alpha =\frac{1}{e}$
$\because 0<\frac{1}{e}<1$ and $2\alpha \in [-2\pi ,2\pi ]$
There will be two values of $2\alpha$ satisfying $cos2\alpha =\frac{1}{e}in[0,2\pi ]$ and two in $[-2\pi 0]$
Therefore, there will be four values of $\alpha in[-2\pi ,2\pi ]$ and correspondingly four values of \beta. Hence, there are four sets of $(\alpha ,\beta ).$