Expand the Trigonometric Expression sin(x+y)sin(x-y)

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

Apply the sum of angles identity.

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

Apply the difference of angles identity.

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

Expand $(\sin (x) \cos (y) + \cos (x) \sin (y)) (\sin (x) \cos (y) - \cos (x) \sin (y))$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify terms.

Combine the opposite terms in $\sin (x) \cos (y) (\sin (x) \cos (y)) + \sin (x) \cos (y) (- \cos (x) \sin (y)) + \cos (x) \sin (y) (\sin (x) \cos (y)) + \cos (x) \sin (y) (- \cos (x) \sin (y))$ .

Reorder the factors in the terms $\sin (x) \cos (y) (- \cos (x) \sin (y))$ and $\cos (x) \sin (y) (\sin (x) \cos (y))$ .

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

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

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

Add $\sin (x) \cos (y) (\sin (x) \cos (y))$ and $0$ .

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

Simplify each term.

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

Raise $\sin (x)$ to the power of $1$ .

$\sin^{1} (x) \sin (x) \cos (y) \cos (y) + \cos (x) \sin (y) (- \cos (x) \sin (y))$

Raise $\sin (x)$ to the power of $1$ .

$\sin^{1} (x) \sin^{1} (x) \cos (y) \cos (y) + \cos (x) \sin (y) (- \cos (x) \sin (y))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\sin (x)}^{1 + 1} \cos (y) \cos (y) + \cos (x) \sin (y) (- \cos (x) \sin (y))$

Add $1$ and $1$ .

$\sin^{2} (x) \cos (y) \cos (y) + \cos (x) \sin (y) (- \cos (x) \sin (y))$

Raise $\cos (y)$ to the power of $1$ .

$\sin^{2} (x) (\cos^{1} (y) \cos (y)) + \cos (x) \sin (y) (- \cos (x) \sin (y))$

Raise $\cos (y)$ to the power of $1$ .

$\sin^{2} (x) (\cos^{1} (y) \cos^{1} (y)) + \cos (x) \sin (y) (- \cos (x) \sin (y))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin^{2} (x) {\cos (y)}^{1 + 1} + \cos (x) \sin (y) (- \cos (x) \sin (y))$

Add $1$ and $1$ .

$\sin^{2} (x) \cos^{2} (y) + \cos (x) \sin (y) (- \cos (x) \sin (y))$

Rewrite using the commutative property of multiplication.

$\sin^{2} (x) \cos^{2} (y) - \cos (x) \sin (y) \cos (x) \sin (y)$

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

Raise $\cos (x)$ to the power of $1$ .

$\sin^{2} (x) \cos^{2} (y) - (\cos^{1} (x) \cos (x)) \sin (y) \sin (y)$

Raise $\cos (x)$ to the power of $1$ .

$\sin^{2} (x) \cos^{2} (y) - (\cos^{1} (x) \cos^{1} (x)) \sin (y) \sin (y)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin^{2} (x) \cos^{2} (y) - {\cos (x)}^{1 + 1} \sin (y) \sin (y)$

Add $1$ and $1$ .

$\sin^{2} (x) \cos^{2} (y) - \cos^{2} (x) \sin (y) \sin (y)$

Multiply $- \cos^{2} (x) \sin (y) \sin (y)$ .

Raise $\sin (y)$ to the power of $1$ .

$\sin^{2} (x) \cos^{2} (y) - \cos^{2} (x) (\sin^{1} (y) \sin (y))$

Raise $\sin (y)$ to the power of $1$ .

$\sin^{2} (x) \cos^{2} (y) - \cos^{2} (x) (\sin^{1} (y) \sin^{1} (y))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin^{2} (x) \cos^{2} (y) - \cos^{2} (x) {\sin (y)}^{1 + 1}$

Add $1$ and $1$ .

$\sin^{2} (x) \cos^{2} (y) - \cos^{2} (x) \sin^{2} (y)$

Expand the Trigonometric Expression sin(x+y)sin(x-y)

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