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)

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

We showed that the two sides could take the same form

\Rightarrow True

cot (- x) sin (x) = - cos (x)

\frac{1 + tan ^{2} ( x )}{1 - tan ^{2} ( x )} = sec (2 x)

sin (3 π - x) = sin (x)

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

sin (3 x) = 4 sin (x) cos^{2} (x) - sin (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