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

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

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

prove

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

True

Solution steps

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

Manipulating right side

sin (4 x)

Rewrite using trig identities

sin (4 x)

= sin (2 \cdot 2 x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (2 x) cos (2 x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 \cdot 2 sin (x) cos (x) cos (2 x)

Simplify

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

Use the Product to Sum identity:

cos (s) sin (t) = \frac{1}{2} (sin (s + t) - sin (s - t))

= 4 cos (x) \frac{1}{2} (sin (2 x + x) - sin (2 x - x))

Simplify

4 cos (x) \frac{1}{2} (sin (2 x + x) - sin (2 x - x)) : 2 cos (x) (sin (3 x) - sin (x))

4 cos (x) \frac{1}{2} (sin (2 x + x) - sin (2 x - x))

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 4 cos ( x ) ( sin ( 2 x + x ) - sin ( 2 x - x ) )}{2}

1 \cdot 4 cos (x) (sin (2 x + x) - sin (2 x - x)) = 4 cos (x) (sin (3 x) - sin (x))

1 \cdot 4 cos (x) (sin (2 x + x) - sin (2 x - x))

Add similar elements:

2 x + x = 3 x

= 1 \cdot 4 cos (x) (sin (3 x) - sin (2 x - x))

Add similar elements:

2 x - x = x

= 1 \cdot 4 cos (x) (sin (3 x) - sin (x))

Multiply the numbers:

1 \cdot 4 = 4

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

= \frac{4 cos ( x ) ( sin ( 3 x ) - sin ( x ) )}{2}

Divide the numbers:

\frac{4}{2} = 2

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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