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Question: If \(\cos(A + B) = \alpha\cos A\cos B + \beta\sin A\sin B,\) then \((\alpha,\beta)\) =...

If cos⁡(A+B)=αcos⁡Acos⁡B+βsin⁡Asin⁡B,\cos(A + B) = \alpha\cos A\cos B + \beta\sin A\sin B, then (α,β)(\alpha,\beta) =

A

(– 1, – 1)

B

(– 1, 1)

C

(1, – 1)

D

(1, 1)

Answer

(1, – 1)

Explanation

Solution

Given cos⁡(A+B)=αcos⁡Acos⁡B+βsin⁡Asin⁡B\cos(A + B) = \alpha\cos A\cos B + \beta\sin A\sin B

But cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A + B) = \cos A\cos B - \sin A\sin B

⇒α=1,β=−1.\Rightarrow \alpha = 1,\beta = - 1.