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

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

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

prove

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

True

Solution steps

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

Manipulating right side

cos (2 x) - cos (2 y)

Rewrite using trig identities

cos (2 x) - cos (2 y)

Use the Sum to Product identity:

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

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

Simplify

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

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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