We have proved the trigonometric expression cos A + cos 2 A + cos 5 A = cos 2 A (1 + 2 cos 3 A) by using C/D formula.
We are given the expression:
cos A + cos 2 A + cos 5 A = cos 2 A (1 + 2 cos 3 A)
We need to prove the expression.
We know that:
Cos C + Cos D = 2 cos ( C + D / 2) cos ( C - D / 2)
Using this identity, we get that:
= cos 5 A + cos A + cos 2 A = cos 2 A (1 + 2 cos 3 A)
= 2 cos ( 5A + A / 2) cos ( 5A - A / 2) + cos 2 A = cos 2 A (1 + 2 cos 3 A)
= 2 cos 3 A cos 2 A + cos 2 A = cos 2 A (1 + 2 cos 3 A)
= cos 2 A ( 2 cos 3A + 1) = cos 2 A (1 + 2 cos 3 A)
= cos 2 A (1 + 2 cos 3 A) = cos 2 A (1 + 2 cos 3 A)
LHS = RHS
Hence proved.
Therefore, we have proved the trigonometric expression cos A + cos 2 A + cos 5 A = cos 2 A (1 + 2 cos 3 A) by using C/D formula.
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