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

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

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

prove

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

True

Solution steps

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

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

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

Simplify

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

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

Multiply fractions:

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

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

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

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

Add similar elements:

2 x - x = x

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

Add similar elements:

2 x + x = 3 x

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

Multiply the numbers:

1 \cdot 4 = 4

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

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

Divide the numbers:

\frac{4}{2} = 2

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

1 - cos^{2} (x) - cos^{2} (x) = - cos (2 x)

sec^{2} (θ) - cos^{2} (θ) = tan (θ) sin (θ)

\frac{1}{sec ( θ ) - tan ( θ )} = sec (θ) tan (θ)

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

16 (\frac{1 - cos ( 2 θ )}{2}) = 8 - 8 cos (2 θ)

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