Algebra Examples

$\tan^{2} (x) - \frac{\sin^{2} (x)}{\cot^{2} (x)}$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Rewrite $\frac{\sin^{2} (x)}{\cot^{2} (x)}$ as ${(\frac{\sin (x)}{\cot (x)})}^{2}$ .

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

Step 1.4

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 1.5

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

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

Step 1.6

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 1.7

Simplify.

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Step 1.7.1

Divide $\sin (x)$ by $1$ .

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

Step 1.7.2

Combine $\sin (x)$ and $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.8

Simplify the numerator.

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Step 1.8.1

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

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

Add $1$ and $1$ .

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

Step 1.9

Apply the product rule to $\frac{\sin^{2} (x)}{\cos (x)}$ .

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

Step 1.10

Multiply the exponents in ${(\sin^{2} (x))}^{2}$ .

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Step 1.10.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.10.2

Multiply $2$ by $2$ .

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

Step 2

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

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

Step 3

Rewrite $\frac{\sin^{4} (x)}{\cos^{2} (x)}$ as ${(\frac{\sin^{2} (x)}{\cos (x)})}^{2}$ .

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

Step 4

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

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

Step 5

Simplify terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

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

Step 5.1.2

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

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

Step 5.1.3

Separate fractions.

$(\tan (x) + \frac{\sin (x)}{1} \cdot \frac{\sin (x)}{\cos (x)}) (\frac{\sin (x)}{\cos (x)} - \frac{\sin^{2} (x)}{\cos (x)})$

Step 5.1.4

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

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

Step 5.1.5

Divide $\sin (x)$ by $1$ .

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

Step 5.2

Simplify each term.

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Step 5.2.1

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

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

Step 5.2.2

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

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

Step 5.2.3

Separate fractions.

$(\tan (x) + \sin (x) \tan (x)) (\tan (x) - (\frac{\sin (x)}{1} \cdot \frac{\sin (x)}{\cos (x)}))$

Step 5.2.4

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

$(\tan (x) + \sin (x) \tan (x)) (\tan (x) - (\frac{\sin (x)}{1} \tan (x)))$

Step 5.2.5

Divide $\sin (x)$ by $1$ .

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

Step 6

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

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 7

Simplify terms.

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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

Add $- \sin (x) \tan (x) \tan (x)$ and $\sin (x) \tan (x) \tan (x)$ .

$\tan (x) \tan (x) + 0 + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Step 7.1.3

Add $\tan (x) \tan (x)$ and $0$ .

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

Step 7.2

Simplify each term.

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Step 7.2.1

Multiply $\tan (x) \tan (x)$ .

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Step 7.2.1.1

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

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

Step 7.2.1.2

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

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

Step 7.2.1.3

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

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

Step 7.2.1.4

Add $1$ and $1$ .

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

Step 7.2.2

Rewrite using the commutative property of multiplication.

$\tan^{2} (x) - \sin (x) \tan (x) \sin (x) \tan (x)$

Step 7.2.3

Multiply $- \sin (x) \tan (x) \sin (x)$ .

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Step 7.2.3.1

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

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

Step 7.2.3.2

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

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

Step 7.2.3.3

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

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

Step 7.2.3.4

Add $1$ and $1$ .

$\tan^{2} (x) - \sin^{2} (x) \tan (x) \tan (x)$

Step 7.2.4

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

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Step 7.2.4.1

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

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

Step 7.2.4.2

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

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

Step 7.2.4.3

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

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

Step 7.2.4.4

Add $1$ and $1$ .

$\tan^{2} (x) - \sin^{2} (x) \tan^{2} (x)$

Step 7.3

Factor $\tan^{2} (x)$ out of $\tan^{2} (x) - \sin^{2} (x) \tan^{2} (x)$ .

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Step 7.3.1

Multiply by $1$ .

$\tan^{2} (x) \cdot 1 - \sin^{2} (x) \tan^{2} (x)$

Step 7.3.2

Factor $\tan^{2} (x)$ out of $- \sin^{2} (x) \tan^{2} (x)$ .

$\tan^{2} (x) \cdot 1 + \tan^{2} (x) (- \sin^{2} (x))$

Step 7.3.3

Factor $\tan^{2} (x)$ out of $\tan^{2} (x) \cdot 1 + \tan^{2} (x) (- \sin^{2} (x))$ .

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

Step 8

Apply pythagorean identity.

$\tan^{2} (x) \cos^{2} (x)$

Step 9

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

${(\frac{\sin (x)}{\cos (x)})}^{2} \cos^{2} (x)$

Step 10

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} \cos^{2} (x)$

Step 11

Cancel the common factor of $\cos^{2} (x)$ .

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Step 11.1

Cancel the common factor.

$\frac{\sin^{2} (x)}{\cos^{2} (x)} \cos^{2} (x)$

Step 11.2

Rewrite the expression.

$\sin^{2} (x)$