Is cos(a+b)-cos(a-b)=-2sin(a)sin(b) ?

The answer to whether cos(a+b)-cos(a-b)=-2sin(a)sin(b) is True

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prove cos(a+b)-cos(a-b)=-2sin(a)sin(b)

prove

cos (a + b) - cos (a - b) = - 2 sin (a) sin (b)

True

Solution steps

cos (a + b) - cos (a - b) = - 2 sin (a) sin (b)

Manipulating right side

- 2 sin (a) sin (b)

Rewrite using trig identities

- 2 sin (a) sin (b)

Use the Product to Sum identity:

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

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

Simplify

- 2 \cdot \frac{1}{2} (cos (a - b) - cos (a + b)) : - cos (a - b) + cos (a + b)

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

Multiply fractions:

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

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

Cancel the common factor:

2

= - (cos (a - b) - cos (a + b)) \cdot 1

Multiply:

(cos (a - b) - cos (a + b)) \cdot 1 = (cos (a - b) - cos (a + b))

= - (cos (a - b) - cos (a + b))

Distribute parentheses

= - (cos (a - b)) - (- cos (a + b))

Apply minus-plus rules

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

= - cos (a - b) + cos (a + b)

= - cos (a - b) + cos (a + b)

= - cos (a - b) + cos (a + b)

= cos (a + b) - cos (a - b)

We showed that the two sides could take the same form

\Rightarrow True

Is cos(a+b)-cos(a-b)=-2sin(a)sin(b) ?
The answer to whether cos(a+b)-cos(a-b)=-2sin(a)sin(b) is True