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Question
If cos(α−β)=1 and cos(α+β)=1e, where α, β∈[−π,π]. Then number of ordered pairs of (α,β) is
If cos(α−β)=1 and cos(α+β)=1e, where α, β∈[−π,π]. Then number of ordered pairs of (α,β) is
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Solution
The correct option is C 4
Since, cos(α−β)=1⇒α−β=nπ,But −2π≤α−β≤2π [ as, α,β∈(−π,π)]∴α−β={0,−2π,2π} ...(i)And α+β={2α,2α±2π}Given, cos(α+β)=1e⇒cos2α=1e<1, which is true for four values of α. as −2π<2α<2π
Since, cos(α−β)=1
⇒α−β=nπ,
But −2π≤α−β≤2π [ as, α,β∈(−π,π)]
∴α−β={0,−2π,2π} ...(i)
And α+β={2α,2α±2π}
Given, cos(α+β)=1e
⇒cos2α=1e<1, which is true for four values of α. as −2π<2α<2π
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