Algebra Examples

$\tan^{2} (θ) + \frac{1}{25} (\cos^{2} (θ) - 1)$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $1$ out of $\tan^{2} (θ) + \frac{1}{25} \cdot (\cos^{2} (θ) - 1)$ .

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Step 1.1.1

Multiply by $1$ .

$1 \tan^{2} (θ) + \frac{1}{25} \cdot (\cos^{2} (θ) - 1)$

Step 1.1.2

Factor $1$ out of $\frac{1}{25} \cdot (\cos^{2} (θ) - 1)$ .

$1 \tan^{2} (θ) + 1 (\frac{1}{25} \cdot (\cos^{2} (θ) - 1))$

Step 1.1.3

Factor $1$ out of $1 \tan^{2} (θ) + 1 (\frac{1}{25} \cdot (\cos^{2} (θ) - 1))$ .

$1 (\tan^{2} (θ) + \frac{1}{25} \cdot (\cos^{2} (θ) - 1))$

Step 1.2

Reorder $\cos^{2} (θ)$ and $- 1$ .

$1 (\tan^{2} (θ) + \frac{1}{25} \cdot (- 1 + \cos^{2} (θ)))$

Step 1.3

Rewrite $- 1$ as $- 1 (1)$ .

$1 (\tan^{2} (θ) + \frac{1}{25} \cdot (- 1 (1) + \cos^{2} (θ)))$

Step 1.4

Factor $- 1$ out of $\cos^{2} (θ)$ .

$1 (\tan^{2} (θ) + \frac{1}{25} \cdot (- 1 (1) - 1 (- \cos^{2} (θ))))$

Step 1.5

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (θ))$ .

$1 (\tan^{2} (θ) + \frac{1}{25} \cdot (- 1 (1 - \cos^{2} (θ))))$

Step 1.6

Rewrite $- 1 (1 - \cos^{2} (θ))$ as $- (1 - \cos^{2} (θ))$ .

$1 (\tan^{2} (θ) + \frac{1}{25} \cdot (- (1 - \cos^{2} (θ))))$

Step 2

Apply pythagorean identity.

$1 (\tan^{2} (θ) + \frac{1}{25} \cdot (- \sin^{2} (θ)))$

Step 3

Simplify terms.

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Step 3.1

Multiply $\tan^{2} (θ) + \frac{1}{25} \cdot (- \sin^{2} (θ))$ by $1$ .

$\tan^{2} (θ) + \frac{1}{25} \cdot (- \sin^{2} (θ))$

Step 3.2

Simplify each term.

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Step 3.2.1

Rewrite $\tan (θ)$ in terms of sines and cosines.

${(\frac{\sin (θ)}{\cos (θ)})}^{2} + \frac{1}{25} \cdot (- \sin^{2} (θ))$

Step 3.2.2

Apply the product rule to $\frac{\sin (θ)}{\cos (θ)}$ .

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)} + \frac{1}{25} \cdot (- \sin^{2} (θ))$

Step 3.2.3

Combine $\frac{1}{25}$ and $\sin^{2} (θ)$ .

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)} - \frac{\sin^{2} (θ)}{25}$

Step 4

Rewrite $\frac{\sin^{2} (θ)}{\cos^{2} (θ)}$ as ${(\frac{\sin (θ)}{\cos (θ)})}^{2}$ .

${(\frac{\sin (θ)}{\cos (θ)})}^{2} - \frac{\sin^{2} (θ)}{25}$

Step 5

Rewrite $\frac{\sin^{2} (θ)}{25}$ as ${(\frac{\sin (θ)}{5})}^{2}$ .

${(\frac{\sin (θ)}{\cos (θ)})}^{2} - {(\frac{\sin (θ)}{5})}^{2}$

Step 6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{\sin (θ)}{\cos (θ)}$ and $b = \frac{\sin (θ)}{5}$ .

$(\frac{\sin (θ)}{\cos (θ)} + \frac{\sin (θ)}{5}) (\frac{\sin (θ)}{\cos (θ)} - \frac{\sin (θ)}{5})$

Step 7

Convert from $\frac{\sin (θ)}{\cos (θ)}$ to $\tan (θ)$ .

$(\tan (θ) + \frac{\sin (θ)}{5}) (\frac{\sin (θ)}{\cos (θ)} - \frac{\sin (θ)}{5})$

Step 8

Convert from $\frac{\sin (θ)}{\cos (θ)}$ to $\tan (θ)$ .

$(\tan (θ) + \frac{\sin (θ)}{5}) (\tan (θ) - \frac{\sin (θ)}{5})$

Step 9

Expand $(\tan (θ) + \frac{\sin (θ)}{5}) (\tan (θ) - \frac{\sin (θ)}{5})$ using the FOIL Method.

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Step 9.1

Apply the distributive property.

$\tan (θ) (\tan (θ) - \frac{\sin (θ)}{5}) + \frac{\sin (θ)}{5} (\tan (θ) - \frac{\sin (θ)}{5})$

Step 9.2

Apply the distributive property.

$\tan (θ) \tan (θ) + \tan (θ) (- \frac{\sin (θ)}{5}) + \frac{\sin (θ)}{5} (\tan (θ) - \frac{\sin (θ)}{5})$

Step 9.3

Apply the distributive property.

$\tan (θ) \tan (θ) + \tan (θ) (- \frac{\sin (θ)}{5}) + \frac{\sin (θ)}{5} \tan (θ) + \frac{\sin (θ)}{5} (- \frac{\sin (θ)}{5})$

Step 10

Simplify terms.

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Step 10.1

Combine the opposite terms in $\tan (θ) \tan (θ) + \tan (θ) (- \frac{\sin (θ)}{5}) + \frac{\sin (θ)}{5} \tan (θ) + \frac{\sin (θ)}{5} (- \frac{\sin (θ)}{5})$ .

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Step 10.1.1

Reorder the factors in the terms $\tan (θ) (- \frac{\sin (θ)}{5})$ and $\frac{\sin (θ)}{5} \tan (θ)$ .

$\tan (θ) \tan (θ) - \tan (θ) \frac{\sin (θ)}{5} + \tan (θ) \frac{\sin (θ)}{5} + \frac{\sin (θ)}{5} (- \frac{\sin (θ)}{5})$

Step 10.1.2

Add $- \tan (θ) \frac{\sin (θ)}{5}$ and $\tan (θ) \frac{\sin (θ)}{5}$ .

$\tan (θ) \tan (θ) + 0 + \frac{\sin (θ)}{5} (- \frac{\sin (θ)}{5})$

Step 10.1.3

Add $\tan (θ) \tan (θ)$ and $0$ .

$\tan (θ) \tan (θ) + \frac{\sin (θ)}{5} (- \frac{\sin (θ)}{5})$

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply $\tan (θ) \tan (θ)$ .

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Step 10.2.1.1

Raise $\tan (θ)$ to the power of $1$ .

$\tan^{1} (θ) \tan (θ) + \frac{\sin (θ)}{5} (- \frac{\sin (θ)}{5})$

Step 10.2.1.2

Raise $\tan (θ)$ to the power of $1$ .

$\tan^{1} (θ) \tan^{1} (θ) + \frac{\sin (θ)}{5} (- \frac{\sin (θ)}{5})$

Step 10.2.1.3

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

${\tan (θ)}^{1 + 1} + \frac{\sin (θ)}{5} (- \frac{\sin (θ)}{5})$

Step 10.2.1.4

Add $1$ and $1$ .

$\tan^{2} (θ) + \frac{\sin (θ)}{5} (- \frac{\sin (θ)}{5})$

Step 10.2.2

Rewrite using the commutative property of multiplication.

$\tan^{2} (θ) - \frac{\sin (θ)}{5} \cdot \frac{\sin (θ)}{5}$

Step 10.2.3

Multiply $- \frac{\sin (θ)}{5} \cdot \frac{\sin (θ)}{5}$ .

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Step 10.2.3.1

Multiply $\frac{\sin (θ)}{5}$ by $\frac{\sin (θ)}{5}$ .

$\tan^{2} (θ) - \frac{\sin (θ) \sin (θ)}{5 \cdot 5}$

Step 10.2.3.2

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

$\tan^{2} (θ) - \frac{\sin^{1} (θ) \sin (θ)}{5 \cdot 5}$

Step 10.2.3.3

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

$\tan^{2} (θ) - \frac{\sin^{1} (θ) \sin^{1} (θ)}{5 \cdot 5}$

Step 10.2.3.4

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

$\tan^{2} (θ) - \frac{{\sin (θ)}^{1 + 1}}{5 \cdot 5}$

Step 10.2.3.5

Add $1$ and $1$ .

$\tan^{2} (θ) - \frac{\sin^{2} (θ)}{5 \cdot 5}$

Step 10.2.3.6

Multiply $5$ by $5$ .

$\tan^{2} (θ) - \frac{\sin^{2} (θ)}{25}$