Is sin(α+β)+sin(α-β)=2sin(α)cos(β) ?

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

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prove sin(α+β)+sin(α-β)=2sin(α)cos(β)

prove

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

True

Solution steps

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

Manipulating right side

2 sin (α) cos (β)

Rewrite using trig identities

2 sin (α) cos (β)

Use the Product to Sum identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= (sin (α + β) + sin (α - β)) \cdot 1

Refine

= sin (α + β) + sin (α - β)

= sin (α + β) + sin (α - β)

= sin (α + β) + sin (α - β)

We showed that the two sides could take the same form

\Rightarrow True

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

2 sin (x) cos (x) = sin (2 x)

cos (x) tan^{2} (x) + cos (x) = sec (x)

1 - sin (x) = cos (x)

(cos (2 x) + sin (2 x))^{2} = 1 + sin (4 x)

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