Is sin(5x)-sin(3x)=2cos(4x)sin(x) ?

The answer to whether sin(5x)-sin(3x)=2cos(4x)sin(x) is True

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prove sin(5x)-sin(3x)=2cos(4x)sin(x)

prove

sin (5 x) - sin (3 x) = 2 cos (4 x) sin (x)

True

Solution steps

sin (5 x) - sin (3 x) = 2 cos (4 x) sin (x)

Manipulating left side

sin (5 x) - sin (3 x)

Rewrite using trig identities

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

Simplify

2 sin (\frac{5 x - 3 x}{2}) cos (\frac{5 x + 3 x}{2}) : 2 sin (x) cos (4 x)

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

\frac{5 x - 3 x}{2} = x

\frac{5 x - 3 x}{2}

Add similar elements:

5 x - 3 x = 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= 2 sin (x) cos (\frac{5 x + 3 x}{2})

\frac{5 x + 3 x}{2} = 4 x

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

Add similar elements:

5 x + 3 x = 8 x

= \frac{8 x}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4 x

= 2 sin (x) cos (4 x)

= 2 sin (x) cos (4 x)

= 2 sin (x) cos (4 x)

= 2 cos (4 x) sin (x)

We showed that the two sides could take the same form

\Rightarrow True

Is sin(5x)-sin(3x)=2cos(4x)sin(x) ?
The answer to whether sin(5x)-sin(3x)=2cos(4x)sin(x) is True