Trigonometry Examples

tan (x) (sin (x) + cot (x) \cdot cos (x))

$\tan (x) (\sin (x) + \cot (x) \cdot \cos (x))$

Step 1

Rewrite

tan (x)

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

\frac{sin (x)}{cos (x)} (sin (x) + cot (x) \cdot cos (x))

$\frac{\sin (x)}{\cos (x)} (\sin (x) + \cot (x) \cdot \cos (x))$

Step 2

Simplify each term.

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Step 2.1

Rewrite

cot (x)

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

\frac{sin (x)}{cos (x)} (sin (x) + \frac{cos (x)}{sin (x)} \cdot cos (x))

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

Step 2.2

Multiply

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

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

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Step 2.2.1

Combine

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

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

cos (x)

$\cos (x)$ .

\frac{sin (x)}{cos (x)} (sin (x) + \frac{cos (x) cos (x)}{sin (x)})

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

Step 2.2.2

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

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

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

Step 2.2.3

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

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

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

Step 2.2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 2.2.5

Add

1

$1$ and

1

$1$ .

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

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

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

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

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

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

Step 3

Apply the distributive property.

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

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

Step 4

Multiply

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

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

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Step 4.1

Combine

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

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

sin (x)

$\sin (x)$ .

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

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

Step 4.2

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

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

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

Step 4.3

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

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

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

Step 4.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 4.5

Add

1

$1$ and

1

$1$ .

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

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

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

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

Step 5

Combine.

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

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

Step 6

Simplify each term.

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Step 6.1

Cancel the common factor of

sin (x)

$\sin (x)$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

Cancel the common factor of

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

$\cos (x)$ .

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Step 6.2.1

Factor

$\cos (x)$ out of

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

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

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Multiply by

$1$ .

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

Step 6.2.2.2

Cancel the common factor.

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

Step 6.2.2.3

Rewrite the expression.

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

Step 6.2.2.4

Divide

$\cos (x)$ by

$1$ .

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

Step 7

Simplify each term.

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Step 7.1

Factor

$\sin (x)$ out of

$\sin^{2} (x)$ .

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

Step 7.2

Separate fractions.

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

Step 7.3

Convert from

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

$\tan (x)$ .

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

Step 7.4

Divide

$\sin (x)$ by

$1$ .

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

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