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Question: If \(\cos(\alpha - \beta) = 1\) and\(\cos(\alpha + \beta) = \frac{1}{e}\), \(- \pi < \alpha,\beta < ...

If cos⁡(α−β)=1\cos(\alpha - \beta) = 1 andcos⁡(α+β)=1e\cos(\alpha + \beta) = \frac{1}{e}, −π<α,β<π- \pi < \alpha,\beta < \pi, then total number of ordered pair of (α,β)(\alpha,\beta) is.

A

0

B

1

C

2

D

4

Answer

4

Explanation

Solution

−2π<α−β<2π- 2\pi < \alpha - \beta < 2\pi

cos⁡(α−β)=1\cos(\alpha - \beta) = 1⇒ α−β=0\alpha - \beta = 0⇒ α=β\alpha = \beta

cos⁡2α=1e\cos 2\alpha = \frac{1}{e}and −2π<2α<2π- 2\pi < 2\alpha < 2\pi

Hence, there will be four solutions.