Is cos(7x)-cos(3x)=-2sin(5x)sin(2x) ?

The answer to whether cos(7x)-cos(3x)=-2sin(5x)sin(2x) is True

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prove cos(7x)-cos(3x)=-2sin(5x)sin(2x)

prove

cos (7 x) - cos (3 x) = - 2 sin (5 x) sin (2 x)

True

Solution steps

cos (7 x) - cos (3 x) = - 2 sin (5 x) sin (2 x)

Manipulating left side

cos (7 x) - cos (3 x)

Rewrite using trig identities

cos (7 x) - cos (3 x)

Use the Sum to Product identity:

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

= - 2 sin (\frac{7 x + 3 x}{2}) sin (\frac{7 x - 3 x}{2})

Simplify

- 2 sin (\frac{7 x + 3 x}{2}) sin (\frac{7 x - 3 x}{2}) : - 2 sin (5 x) sin (2 x)

- 2 sin (\frac{7 x + 3 x}{2}) sin (\frac{7 x - 3 x}{2})

\frac{7 x + 3 x}{2} = 5 x

\frac{7 x + 3 x}{2}

Add similar elements:

7 x + 3 x = 10 x

= \frac{10 x}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 x

= - 2 sin (5 x) sin (\frac{7 x - 3 x}{2})

\frac{7 x - 3 x}{2} = 2 x

\frac{7 x - 3 x}{2}

Add similar elements:

7 x - 3 x = 4 x

= \frac{4 x}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 x

= - 2 sin (5 x) sin (2 x)

= - 2 sin (5 x) sin (2 x)

= - 2 sin (5 x) sin (2 x)

We showed that the two sides could take the same form

\Rightarrow True

Is cos(7x)-cos(3x)=-2sin(5x)sin(2x) ?
The answer to whether cos(7x)-cos(3x)=-2sin(5x)sin(2x) is True