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

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

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

prove

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

True

Solution steps

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

Manipulating right side

2 sin (x) sin (y)

Rewrite using trig identities

2 sin (x) sin (y)

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 (x - y) - cos (x + y))

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Refine

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

tan (θ) csc (θ) = sec (θ)

(sec (θ) + tan (θ)) (sec (θ) - tan (θ)) = 1

\frac{cos ( u ) sec ( u )}{tan ( u )} = cot (u)

sec^{2} (x) (1 - sin^{2} (x)) = 1

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

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