Trigonometry Examples

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

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

Step 1

Start on the left side.

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

$\sin (x + y) - \sin (x - y)$

Step 2

Apply the sum of angles identity.

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

$\sin (x) \cos (y) + \cos (x) \sin (y) - \sin (x - y)$

Step 3

Apply the sum of angles identity.

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

$\sin (x) \cos (y) + \cos (x) \sin (y) - (\sin (x) \cos (- y) + \cos (x) \sin (- y))$

Step 4

Simplify the expression.

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Step 4.1

Simplify each term.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Since

cos (- y)

$\cos (- y)$ is an even function, rewrite

cos (- y)

$\cos (- y)$ as

cos (y)

$\cos (y)$ .

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

$\sin (x) \cos (y) + \cos (x) \sin (y) - (\sin (x) \cos (y) + \cos (x) \sin (- y))$

Step 4.1.1.2

Since

sin (- y)

$\sin (- y)$ is an odd function, rewrite

sin (- y)

$\sin (- y)$ as

- sin (y)

$- \sin (y)$ .

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

$\sin (x) \cos (y) + \cos (x) \sin (y) - (\sin (x) \cos (y) + \cos (x) (- \sin (y)))$

Step 4.1.1.3

Rewrite using the commutative property of multiplication.

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

$\sin (x) \cos (y) + \cos (x) \sin (y) - (\sin (x) \cos (y) - \cos (x) \sin (y))$

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

$\sin (x) \cos (y) + \cos (x) \sin (y) - (\sin (x) \cos (y) - \cos (x) \sin (y))$

Step 4.1.2

Apply the distributive property.

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

$\sin (x) \cos (y) + \cos (x) \sin (y) - (\sin (x) \cos (y)) - (- \cos (x) \sin (y))$

Step 4.1.3

Multiply

- (- cos (x) sin (y))

$- (- \cos (x) \sin (y))$ .

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Step 4.1.3.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

sin (x) cos (y) + cos (x) sin (y) - sin (x) cos (y) + 1 (cos (x) sin (y))

$\sin (x) \cos (y) + \cos (x) \sin (y) - \sin (x) \cos (y) + 1 (\cos (x) \sin (y))$

Step 4.1.3.2

Multiply

cos (x)

$\cos (x)$ by

1

$1$ .

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

$\sin (x) \cos (y) + \cos (x) \sin (y) - \sin (x) \cos (y) + \cos (x) \sin (y)$

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

$\sin (x) \cos (y) + \cos (x) \sin (y) - \sin (x) \cos (y) + \cos (x) \sin (y)$

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

$\sin (x) \cos (y) + \cos (x) \sin (y) - \sin (x) \cos (y) + \cos (x) \sin (y)$

Step 4.2

Combine the opposite terms in

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

$\sin (x) \cos (y) + \cos (x) \sin (y) - \sin (x) \cos (y) + \cos (x) \sin (y)$ .

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Step 4.2.1

Subtract

sin (x) cos (y)

$\sin (x) \cos (y)$ from

sin (x) cos (y)

$\sin (x) \cos (y)$ .

cos (x) sin (y) + 0 + cos (x) sin (y)

$\cos (x) \sin (y) + 0 + \cos (x) \sin (y)$

Step 4.2.2

Add

cos (x) sin (y)

$\cos (x) \sin (y)$ and

0

$0$ .

cos (x) sin (y) + cos (x) sin (y)

$\cos (x) \sin (y) + \cos (x) \sin (y)$

cos (x) sin (y) + cos (x) sin (y)

$\cos (x) \sin (y) + \cos (x) \sin (y)$

Step 4.3

Add

cos (x) sin (y)

$\cos (x) \sin (y)$ and

cos (x) sin (y)

$\cos (x) \sin (y)$ .

2 cos (x) sin (y)

$2 \cos (x) \sin (y)$

2 cos (x) sin (y)

$2 \cos (x) \sin (y)$

Step 5

Because the two sides have been shown to be equivalent, the equation is an identity.

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

$\sin (x + y) - \sin (x - y) = 2 \cos (x) \sin (y)$ is an identity