Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Expand $(\sin (x) + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \frac{\sin (x)}{\cos (x)})$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

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

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Step 1.4.1.1.1

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

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

Step 1.4.1.1.2

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

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

Step 1.4.1.1.3

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

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

Step 1.4.1.1.4

Add $1$ and $1$ .

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

Step 1.4.1.2

Multiply $\sin (x) \frac{\sin (x)}{\cos (x)}$ .

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Step 1.4.1.2.1

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

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

Step 1.4.1.2.2

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

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

Step 1.4.1.2.3

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

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

Step 1.4.1.2.4

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

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

Step 1.4.1.2.5

Add $1$ and $1$ .

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

Step 1.4.1.3

Multiply $\frac{\sin (x)}{\cos (x)} \sin (x)$ .

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Step 1.4.1.3.1

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

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

Step 1.4.1.3.2

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

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

Step 1.4.1.3.3

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

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

Step 1.4.1.3.4

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

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

Step 1.4.1.3.5

Add $1$ and $1$ .

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

Step 1.4.1.4

Multiply $\frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Step 1.4.1.4.1

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.4.1.4.2

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

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

Step 1.4.1.4.3

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

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

Step 1.4.1.4.4

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

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

Step 1.4.1.4.5

Add $1$ and $1$ .

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

Step 1.4.1.4.6

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

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

Step 1.4.1.4.7

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

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

Step 1.4.1.4.8

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

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

Step 1.4.1.4.9

Add $1$ and $1$ .

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

Step 1.4.2

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 1.8

One to any power is one.

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

Step 1.9

Rewrite $\sin^{2} (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)} + \cos^{2} (x) - \frac{1}{\cos^{2} (x)}$ in a factored form.

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Step 1.9.1

Let $u = \frac{2 \sin^{2} (x)}{\cos (x)}$ . Substitute $u$ for all occurrences of $\frac{2 \sin^{2} (x)}{\cos (x)}$ .

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Step 1.9.1.1

Subtract $1$ from $1$ .

$\frac{u + 0}{\tan (x)}$

Step 1.9.1.2

Add $u$ and $0$ .

$\frac{u}{\tan (x)}$

Step 1.9.2

Replace all occurrences of $u$ with $\frac{2 \sin^{2} (x)}{\cos (x)}$ .

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

Step 2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Combine.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $\sin (x)$ out of $\cos (x) \sin (x)$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 6

Cancel the common factor of $\cos (x)$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Divide $2 \sin (x)$ by $1$ .

$2 \sin (x)$