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Answer:
true
Step-by-step explanation:
1/2 (sin(x+y)+sin(x-y))
= 1/2 (sin x cos Y + cos x sin y + sin x cos y - cos x sin y)
= 1/2 * 2 sin x cos y
= sin x cos y
The expression is true.
The trigonometric identity 1/2 [sin (x + y)+sin (x - y)] = sin x cos y holds true, using the sum and difference identities for sin x.
The sum and difference identities for trigonometric ratios are as follows:-
In the question, we are asked to add the sum and difference identities for sin x to derive 1/2 [sin (x + y)+sin (x - y)] = sin x cos y.
Going by the left-hand side of the identity:-
1/2 [sin (x + y) + sin (x - y)],
Putting sin (x + y) = sin x cos y + cos x sin y, and sin (x - y) = sin x cos y - cos x sin y, we get:
= 1/2 [(sin x cos y + cos x sin y) + (sin x cos y - cos x sin y)]
= 1/2 [sin x cos y + cos x sin y + sin x cos y - cos x sin y]
= 1/2 [ 2 sin x cos y] {Since, cos x sin y is cancelled}
= sin x cos y
= The right-hand side of the identity.
Hence, the identity is derived.
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