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Question
Prove the following:
sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x
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Solution
L.H.S.
= sin (n + 1)x sin (n + 2) x + cos (n + 1)x cos (n + 2)x
Let (n + 2)x = A, (n + 1) x = B
= sin B sin A + cos B cos A
= cos A cos B + sin A sin B
= cos (A – B) = cos [(n + 2) x – (n + 1)x]
[∵ By keeping the values of A and B]
= cos (nx + 2x – nx –x)
= cos x = R.H.S.
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