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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sin (3 x) + sin (x)

Rewrite using trig identities

sin (3 x) + sin (x)

Use the Sum to Product identity:

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

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

Simplify

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

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

= 2 sin (2 x) cos (x)

= 2 sin (2 x) cos (x)

We showed that the two sides could take the same form

\Rightarrow True

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