cos = 1 and cos = 1 e , where . Pairs of which satisfy both the equations is/are [2005]

Question

$\cos (α - β) = 1$ and $\cos (α + β) = \frac{1}{e},$ where $α, β \in [- π, π]$ . Pairs of $α, β$ which satisfy both the equations is/are [2005]

Smart Achievers · Accepted Answer

(4)

$Given: \cos (α - β) = 1 and \cos (α + β) = \frac{1}{e}, where α, β \in [- π, π]$

$Now, \cos (α - β) = 1 \Rightarrow α - β = 0 \Rightarrow α = β$

$and \cos (α + β) = \frac{1}{e} \Rightarrow \cos 2 α = \frac{1}{e}$

$∵$ $0 < \frac{1}{e} < 1$

$Now 2 α \in [- 2 π, 2 π]$

$\Rightarrow There will be two values of 2 α in [- 2 π, 0] satisfying \cos 2 α = \frac{1}{e} and two values in [0, 2 π] .$

$\Rightarrow There will be four values of α in [- π, π] and correspondingly four values of β . Hence there are four sets of (α, β) .$

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