Precalculus Examples

csc (x) - cos (x) cot (x)

$\csc (x) - \cos (x) \cot (x)$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite

csc (x)

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

\frac{1}{sin (x)} - cos (x) cot (x)

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

Step 1.1.2

Rewrite

cot (x)

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

\frac{1}{sin (x)} - cos (x) \frac{cos (x)}{sin (x)}

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

Step 1.1.3

Multiply

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

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

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Step 1.1.3.1

Combine

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

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

cos (x)

$\cos (x)$ .

\frac{1}{sin (x)} - \frac{cos (x) cos (x)}{sin (x)}

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

Step 1.1.3.2

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

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

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

Step 1.1.3.3

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

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

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

Step 1.1.3.4

Use the power rule

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

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

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

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

Step 1.1.3.5

Add

1

$1$ and

1

$1$ .

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

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

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

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

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

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

Step 1.2

Combine the numerators over the common denominator.

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

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

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

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

Step 2

Apply pythagorean identity.

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

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

Step 3

Cancel the common factor of

{sin}^{2} (x)

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

sin (x)

$\sin (x)$ .

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Step 3.1

Factor

sin (x)

$\sin (x)$ out of

{sin}^{2} (x)

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

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Multiply by

1

$1$ .

\frac{sin (x) sin (x)}{sin (x) \cdot 1}

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

$\frac{\sin (x)}{1}$

Step 3.2.4

Divide

$\sin (x)$ by

$1$ .

$\sin (x)$