Basic Math Examples

${(\cos (A) - 2)}^{2} - 4 \sin^{2} (A)$

Step 1

Rewrite $4 \sin^{2} (A)$ as ${(2 \sin (A))}^{2}$ .

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

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (A) - 2$ and $b = 2 \sin (A)$ .

$(\cos (A) - 2 + 2 \sin (A)) (\cos (A) - 2 - (2 \sin (A)))$

Step 3

Multiply $2$ by $- 1$ .

$(\cos (A) - 2 + 2 \sin (A)) (\cos (A) - 2 - 2 \sin (A))$

Step 4

Expand $(\cos (A) - 2 + 2 \sin (A)) (\cos (A) - 2 - 2 \sin (A))$ by multiplying each term in the first expression by each term in the second expression.

$\cos (A) \cos (A) + \cos (A) \cdot - 2 + \cos (A) (- 2 \sin (A)) - 2 \cos (A) - 2 \cdot - 2 - 2 (- 2 \sin (A)) + 2 \sin (A) \cos (A) + 2 \sin (A) \cdot - 2 + 2 \sin (A) (- 2 \sin (A))$

Step 5

Simplify terms.

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Step 5.1

Combine the opposite terms in $\cos (A) \cos (A) + \cos (A) \cdot - 2 + \cos (A) (- 2 \sin (A)) - 2 \cos (A) - 2 \cdot - 2 - 2 (- 2 \sin (A)) + 2 \sin (A) \cos (A) + 2 \sin (A) \cdot - 2 + 2 \sin (A) (- 2 \sin (A))$ .

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Step 5.1.1

Reorder the factors in the terms $\cos (A) (- 2 \sin (A))$ and $2 \sin (A) \cos (A)$ .

$\cos (A) \cos (A) + \cos (A) \cdot - 2 - 2 \cos (A) \sin (A) - 2 \cos (A) - 2 \cdot - 2 - 2 (- 2 \sin (A)) + 2 \cos (A) \sin (A) + 2 \sin (A) \cdot - 2 + 2 \sin (A) (- 2 \sin (A))$

Step 5.1.2

Add $- 2 \cos (A) \sin (A)$ and $2 \cos (A) \sin (A)$ .

$\cos (A) \cos (A) + \cos (A) \cdot - 2 + 0 - 2 \cos (A) - 2 \cdot - 2 - 2 (- 2 \sin (A)) + 2 \sin (A) \cdot - 2 + 2 \sin (A) (- 2 \sin (A))$

Step 5.1.3

Add $\cos (A) \cos (A) + \cos (A) \cdot - 2$ and $0$ .

$\cos (A) \cos (A) + \cos (A) \cdot - 2 - 2 \cos (A) - 2 \cdot - 2 - 2 (- 2 \sin (A)) + 2 \sin (A) \cdot - 2 + 2 \sin (A) (- 2 \sin (A))$

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply $\cos (A) \cos (A)$ .

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Step 5.2.1.1

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

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

Step 5.2.1.2

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

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

Step 5.2.1.3

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

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

Step 5.2.1.4

Add $1$ and $1$ .

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

Step 5.2.2

Move $- 2$ to the left of $\cos (A)$ .

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

Step 5.2.3

Multiply $- 2$ by $- 2$ .

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 - 2 (- 2 \sin (A)) + 2 \sin (A) \cdot - 2 + 2 \sin (A) (- 2 \sin (A))$

Step 5.2.4

Multiply $- 2$ by $- 2$ .

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 + 4 \sin (A) + 2 \sin (A) \cdot - 2 + 2 \sin (A) (- 2 \sin (A))$

Step 5.2.5

Multiply $- 2$ by $2$ .

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 + 4 \sin (A) - 4 \sin (A) + 2 \sin (A) (- 2 \sin (A))$

Step 5.2.6

Multiply $2 \sin (A) (- 2 \sin (A))$ .

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Step 5.2.6.1

Multiply $- 2$ by $2$ .

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 + 4 \sin (A) - 4 \sin (A) - 4 \sin (A) \sin (A)$

Step 5.2.6.2

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

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 + 4 \sin (A) - 4 \sin (A) - 4 (\sin^{1} (A) \sin (A))$

Step 5.2.6.3

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

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 + 4 \sin (A) - 4 \sin (A) - 4 (\sin^{1} (A) \sin^{1} (A))$

Step 5.2.6.4

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

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 + 4 \sin (A) - 4 \sin (A) - 4 {\sin (A)}^{1 + 1}$

Step 5.2.6.5

Add $1$ and $1$ .

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 + 4 \sin (A) - 4 \sin (A) - 4 \sin^{2} (A)$

Step 5.3

Simplify terms.

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Step 5.3.1

Combine the opposite terms in $\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 + 4 \sin (A) - 4 \sin (A) - 4 \sin^{2} (A)$ .

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Step 5.3.1.1

Subtract $4 \sin (A)$ from $4 \sin (A)$ .

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 + 0 - 4 \sin^{2} (A)$

Step 5.3.1.2

Add $\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4$ and $0$ .

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 - 4 \sin^{2} (A)$

Step 5.3.2

Factor $4$ out of $4$ .

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 (1) - 4 \sin^{2} (A)$

Step 5.3.3

Factor $4$ out of $- 4 \sin^{2} (A)$ .

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 (1) + 4 (- \sin^{2} (A))$

Step 5.3.4

Factor $4$ out of $4 (1) + 4 (- \sin^{2} (A))$ .

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 (1 - \sin^{2} (A))$

Step 6

Apply pythagorean identity.

$\cos^{2} (A) - 2 \cos (A) - 2 \cos (A) + 4 \cos^{2} (A)$

Step 7

Simplify by adding terms.

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Step 7.1

Add $\cos^{2} (A)$ and $4 \cos^{2} (A)$ .

$5 \cos^{2} (A) - 2 \cos (A) - 2 \cos (A)$

Step 7.2

Subtract $2 \cos (A)$ from $- 2 \cos (A)$ .

$5 \cos^{2} (A) - 4 \cos (A)$