Is sin(7x)+sin(x)=2cos(3x)sin(4x) ?

The answer to whether sin(7x)+sin(x)=2cos(3x)sin(4x) is True

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prove sin(7x)+sin(x)=2cos(3x)sin(4x)

prove

sin (7 x) + sin (x) = 2 cos (3 x) sin (4 x)

True

Solution steps

sin (7 x) + sin (x) = 2 cos (3 x) sin (4 x)

Manipulating left side

sin (7 x) + sin (x)

Rewrite using trig identities

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

Simplify

2 sin (\frac{7 x + x}{2}) cos (\frac{7 x - x}{2}) : 2 sin (4 x) cos (3 x)

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

\frac{7 x + x}{2} = 4 x

\frac{7 x + x}{2}

Add similar elements:

7 x + x = 8 x

= \frac{8 x}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4 x

= 2 sin (4 x) cos (\frac{7 x - x}{2})

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

\frac{7 x - x}{2}

Add similar elements:

7 x - x = 6 x

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

Is sin(7x)+sin(x)=2cos(3x)sin(4x) ?
The answer to whether sin(7x)+sin(x)=2cos(3x)sin(4x) is True