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Mathematics Question on Trigonometric Functions

The number of ordered pairs (α,β),(\alpha, \beta), where α,β∈(−π,π)\alpha, \beta \in(-\pi, \pi) satisfying cos (α−β)=1andcos(α+β)=1e(\alpha-\beta)=1 and cos (\alpha+\beta)=\frac{1}{e} is

A

0

B

1

C

2

D

4

Answer

4

Explanation

Solution

Since,\hspace25mm cos(\alpha-\beta)=1
\Rightarrow \hspace25mm \alpha-\beta=2n\pi
But \hspace25mm -2\pi < \alpha-\beta < 2\pi [as \alpha, \beta \in (-\pi, \pi)]
\therefore \hspace25mm \alpha-\beta=0 \hspace30mm ...(i)
Given, \hspace25mm co (\alpha+\beta)=\frac{1}{e}
⇒cos2α=1e<1,whichistrueforfourvaluesofα.\Rightarrow cos 2\alpha=\frac{1}{e} < 1, which is true for four values of \alpha.
\hspace25mm [as -2\pi < 2\alpha < 2\pi]