Is sin(x+y)+sin(x-y)=2sin(x)cos(y) ?

The answer to whether sin(x+y)+sin(x-y)=2sin(x)cos(y) is True

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prove sin(x+y)+sin(x-y)=2sin(x)cos(y)

prove

sin (x + y) + sin (x - y) = 2 sin (x) cos (y)

True

Solution steps

sin (x + y) + sin (x - y) = 2 sin (x) cos (y)

Manipulating right side

2 sin (x) cos (y)

Rewrite using trig identities

2 sin (x) cos (y)

Use the Product to Sum identity:

sin (s) cos (t) = \frac{1}{2} (sin (s + t) + sin (s - t))

= 2 \cdot \frac{1}{2} (sin (x + y) + sin (x - y))

Simplify

2 \cdot \frac{1}{2} (sin (x + y) + sin (x - y)) : sin (x + y) + sin (x - y)

2 \cdot \frac{1}{2} (sin (x + y) + sin (x - y))

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2} (sin (x + y) + sin (x - y))

Cancel the common factor:

2

= (sin (x + y) + sin (x - y)) \cdot 1

Refine

= sin (x + y) + sin (x - y)

= sin (x + y) + sin (x - y)

= sin (x + y) + sin (x - y)

We showed that the two sides could take the same form

\Rightarrow True

p ro v e cos (2 θ) = cos^{2} (θ) - sin^{2} (θ)

p ro v e sin (3 θ) = 3 sin (θ) - 4 sin^{3} (θ)

p ro v e sin^{4} (t) - cos^{4} (t) = 1 - 2 cos^{2} (t)

p ro v e (1 - cos^{2} (x)) (1 + cot^{2} (x)) = 1

p ro v e \frac{1 - cos ( - x )}{sec ( - x ) - 1} = cos (x)

Is sin(x+y)+sin(x-y)=2sin(x)cos(y) ?
The answer to whether sin(x+y)+sin(x-y)=2sin(x)cos(y) is True