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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sin (2 x) + sin (2 y)

Rewrite using trig identities

sin (2 x) + sin (2 y)

Use the Sum to Product identity:

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

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

Simplify

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

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

\frac{2 x + 2 y}{2} = x + y

\frac{2 x + 2 y}{2}

Factor out common term

2

= \frac{2 ( x + y )}{2}

Divide the numbers:

\frac{2}{2} = 1

= x + y

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

\frac{2 x - 2 y}{2} = x - y

\frac{2 x - 2 y}{2}

Factor out common term

2

= \frac{2 ( x - y )}{2}

Divide the numbers:

\frac{2}{2} = 1

= x - y

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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cot (x - \frac{π}{2}) = - tan (x)

\frac{sin ( 5 x ) + sin ( 11 x )}{cos ( 5 x ) + cos ( 11 x )} = tan (8 x)

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