Is cos(α-β)-cos(α+β)=2sin(α)sin(β) ?

The answer to whether cos(α-β)-cos(α+β)=2sin(α)sin(β) is True

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prove cos(α-β)-cos(α+β)=2sin(α)sin(β)

prove

cos (α - β) - cos (α + β) = 2 sin (α) sin (β)

True

Solution steps

cos (α - β) - cos (α + β) = 2 sin (α) sin (β)

Manipulating right side

2 sin (α) sin (β)

Rewrite using trig identities

2 sin (α) sin (β)

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 (α - β) - cos (α + β))

Simplify

2 \cdot \frac{1}{2} (cos (α - β) - cos (α + β)) : cos (α - β) - cos (α + β)

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

Multiply fractions:

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

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

Cancel the common factor:

2

= (cos (α - β) - cos (α + β)) \cdot 1

Refine

= cos (α - β) - cos (α + β)

= cos (α - β) - cos (α + β)

= cos (α - β) - cos (α + β)

We showed that the two sides could take the same form

\Rightarrow True

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