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

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

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

prove

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

True

Solution steps

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

Manipulating left side

cos (3 x) - cos (x)

Rewrite using trig identities

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

Simplify

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

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

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

\frac{3 x + x}{2}

Add similar elements:

3 x + x = 4 x

= \frac{4 x}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 x

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

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

\frac{3 x - x}{2}

Add similar elements:

3 x - x = 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

sec (\frac{π}{2} - y) = csc (y)

cos (2 x - \frac{π}{2}) = cos (\frac{π}{2} - 2 x)

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

cos (x) \cdot csc (x) \cdot tan (x) = 1

- 2 sin (2 (x + 45)) = 2 cos (2 x)

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