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

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

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

prove

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

True

Solution steps

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

Manipulating right side

2 sin (a) cos (b)

Rewrite using trig identities

2 sin (a) cos (b)

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 (a + b) + sin (a - b))

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Refine

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

\frac{1 + tan ^{2} ( x )}{2 tan ( x )} = csc (2 x)

cos (3 θ) = cos^{3} (θ) - 3 cos (θ) sin^{2} (θ)

\frac{tan ^{2} ( x )}{sec ( x )} = sin (x) tan (x)

cot (x) (1 - cos (2 x)) = sin (2 x)

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