Math  /  Trigonometry

Question1. 2cosxcosy=cos(x+y)+cos(xy)2 \cos x \cos y=\cos (x+y)+\cos (x-y)

Studdy Solution
Verify the given identity using the sum-to-product identities. We start with the right-hand side of the given identity:
cos(x+y)+cos(xy) \cos (x+y) + \cos (x-y)
Using the sum-to-product identity:
cos(x+y)+cos(xy)=2cos((x+y)+(xy)2)cos((x+y)(xy)2) \cos (x+y) + \cos (x-y) = 2 \cos \left( \frac{(x+y) + (x-y)}{2} \right) \cos \left( \frac{(x+y) - (x-y)}{2} \right)
Simplify the expressions inside the cosines:
=2cos(2x2)cos(2y2) = 2 \cos \left( \frac{2x}{2} \right) \cos \left( \frac{2y}{2} \right) =2cosxcosy = 2 \cos x \cos y
This matches the left-hand side of the given identity:
2cosxcosy=cos(x+y)+cos(xy) 2 \cos x \cos y = \cos (x+y) + \cos (x-y)
The identity is verified as true.

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