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Question
Prove the following:
sin [(n + 1)A]. sin [(n + 2)A] + cos [(n + 1)A]. cos [(n + 2)A] = cos A
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Solution
L.H.S. = sin [(n + 1)A]. sin [(n + 2)A] + cos [(n + 1)A]. cos [(n + 2)A]
= cos [(n + 2)A]. cos [(n + 1)A] + sin [(n + 2)A]. sin [(n + 1)A]
Let (n + 2)A = a and (n + 1)A = b .......(i)
∴ L.H.S. = cos a. cos b + sin a. sin b
= cos (a − b)
= cos [(n + 2)A − (n + 1)A] .....[From (i)]
= cos [(n + 2 − n − 1)A]
= cos A
= R.H.S.
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