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

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

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

prove

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

True

Solution steps

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

Manipulating left side

cos (2 x - x) - 2 sin (x) sin (2 x)

Rewrite using trig identities

cos (2 x - x) - 2 sin (x) sin (2 x)

Use the Product to Sum identity:

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

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

Simplify

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

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

cos (2 x - x) = cos (x)

cos (2 x - x)

Add similar elements:

2 x - x = x

= cos (x)

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 1 \cdot (cos (x - 2 x) - cos (x + 2 x))

Add similar elements:

x - 2 x = - x

= 1 \cdot (cos (- x) - cos (x + 2 x))

Add similar elements:

x + 2 x = 3 x

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

Refine

= cos (- x) - cos (3 x)

Use the negative angle identity:

cos (- x) = cos (x)

= cos (x) - cos (3 x)

= cos (x) - (cos (x) - cos (3 x))

- (cos (x) - cos (3 x)) : - cos (x) + cos (3 x)

- (cos (x) - cos (3 x))

Distribute parentheses

= - (cos (x)) - (- cos (3 x))

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - cos (x) + cos (3 x)

= cos (x) - cos (x) + cos (3 x)

Add similar elements:

cos (x) - cos (x) = 0

= cos (3 x)

= cos (3 x)

= cos (3 x)

We showed that the two sides could take the same form

\Rightarrow True

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