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

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

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

prove

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

True

Solution steps

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

Manipulating left side

2 cos (x) sin (y)

Rewrite using trig identities

2 cos (x) sin (y)

Use the Product to Sum identity:

cos (s) sin (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

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