Is sin(x-y)+sin(x+y)=2cos(y)sin(x) ?

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

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prove sin(x-y)+sin(x+y)=2cos(y)sin(x)

prove

sin (x - y) + sin (x + y) = 2 cos (y) sin (x)

True

Solution steps

sin (x - y) + sin (x + y) = 2 cos (y) sin (x)

Manipulating right side

2 cos (y) sin (x)

Rewrite using trig identities

2 cos (y) sin (x)

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 (x + y) + sin (x - y))

Simplify

2 \cdot \frac{1}{2} (sin (x + y) + sin (x - y)) : sin (x + y) + sin (x - y)

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

Multiply fractions:

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

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

Cancel the common factor:

2

= (sin (x + y) + sin (x - y)) \cdot 1

Refine

= sin (x + y) + sin (x - y)

= sin (x + y) + sin (x - y)

= sin (x + y) + sin (x - y)

= sin (x - y) + sin (x + y)

We showed that the two sides could take the same form

\Rightarrow True

(csc^{2} (x) - 1) sin (x) = cot^{2} (x) sin (x)

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

\frac{3 sin ^{2} ( t )}{tan ^{2} ( t )} = 3 cos^{2} (t)

\frac{1}{tan ^{2} ( θ )} = 1 + cot^{2} (θ)

cos (\frac{4 π}{3}) = - \frac{1}{2}

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