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

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

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

prove

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

True

Solution steps

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

Manipulating left side

4 sin (3 x) cos (4 x)

Rewrite using trig identities

4 sin (3 x) cos (4 x)

Use the Product to Sum identity:

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

= 4 \cdot \frac{1}{2} (sin (4 x + 3 x) - sin (4 x - 3 x))

Simplify

4 \cdot \frac{1}{2} (sin (4 x + 3 x) - sin (4 x - 3 x)) : 2 (sin (7 x) - sin (x))

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

Multiply fractions:

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

= \frac{1 \cdot 4 ( sin ( 4 x + 3 x ) - sin ( 4 x - 3 x ) )}{2}

1 \cdot 4 (sin (4 x + 3 x) - sin (4 x - 3 x)) = 4 (sin (7 x) - sin (x))

1 \cdot 4 (sin (4 x + 3 x) - sin (4 x - 3 x))

Add similar elements:

4 x + 3 x = 7 x

= 1 \cdot 4 (sin (7 x) - sin (4 x - 3 x))

Add similar elements:

4 x - 3 x = x

= 1 \cdot 4 (sin (7 x) - sin (x))

Multiply the numbers:

1 \cdot 4 = 4

= 4 (sin (7 x) - sin (x))

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

Divide the numbers:

\frac{4}{2} = 2

= 2 (sin (7 x) - sin (x))

= 2 (sin (7 x) - sin (x))

= 2 (sin (7 x) - sin (x))

Manipulating right side

2 sin (7 x) - 2 sin (x)

Factor

2 sin (7 x) - 2 sin (x) : 2 (sin (7 x) - sin (x))

2 sin (7 x) - 2 sin (x)

Factor out common term

2

= 2 (sin (7 x) - sin (x))

= (sin (7 x) - sin (x)) \cdot 2

We showed that the two sides could take the same form

\Rightarrow True

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