Trigonometry Examples

${(\sin (t) - \cos (t))}^{2}$

Step 1

Rewrite ${(\sin (t) - \cos (t))}^{2}$ as $(\sin (t) - \cos (t)) (\sin (t) - \cos (t))$ .

$(\sin (t) - \cos (t)) (\sin (t) - \cos (t))$

Step 2

Expand $(\sin (t) - \cos (t)) (\sin (t) - \cos (t))$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sin (t) (\sin (t) - \cos (t)) - \cos (t) (\sin (t) - \cos (t))$

Step 2.2

Apply the distributive property.

$\sin (t) \sin (t) + \sin (t) (- \cos (t)) - \cos (t) (\sin (t) - \cos (t))$

Step 2.3

Apply the distributive property.

$\sin (t) \sin (t) + \sin (t) (- \cos (t)) - \cos (t) \sin (t) - \cos (t) (- \cos (t))$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\sin (t) \sin (t)$ .

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

Add $1$ and $1$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

$\sin^{2} (t) - \sin (t) \cos (t) - \cos (t) \sin (t) - \cos (t) (- \cos (t))$

Step 3.1.3

Multiply $- \cos (t) (- \cos (t))$ .

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Step 3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.3.2

Multiply $\cos (t)$ by $1$ .

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

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

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

Step 3.1.3.6

Add $1$ and $1$ .

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

Step 3.2

Reorder the factors of $- \sin (t) \cos (t)$ .

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

Step 3.3

Subtract $\cos (t) \sin (t)$ from $- \cos (t) \sin (t)$ .

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

Step 4

Move $\cos^{2} (t)$ .

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

Step 5

Apply pythagorean identity.

$1 - 2 \cos (t) \sin (t)$