Precalculus Examples

$(\csc (x) - \cot (x)) (\sec (x) + 1)$

Step 1

Expand $(\csc (x) - \cot (x)) (\sec (x) + 1)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\csc (x) (\sec (x) + 1) - \cot (x) (\sec (x) + 1)$

Step 1.2

Apply the distributive property.

$\csc (x) \sec (x) + \csc (x) \cdot 1 - \cot (x) (\sec (x) + 1)$

Step 1.3

Apply the distributive property.

$\csc (x) \sec (x) + \csc (x) \cdot 1 - \cot (x) \sec (x) - \cot (x) \cdot 1$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $\csc (x)$ by $1$ .

$\csc (x) \sec (x) + \csc (x) - \cot (x) \sec (x) - \cot (x) \cdot 1$

Step 2.1.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 2.1.2.1

Add parentheses.

$\csc (x) \sec (x) + \csc (x) - (\cot (x) \sec (x)) - \cot (x) \cdot 1$

Step 2.1.2.2

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

$\csc (x) \sec (x) + \csc (x) - (\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos (x)}) - \cot (x) \cdot 1$

Step 2.1.2.3

Cancel the common factors.

$\csc (x) \sec (x) + \csc (x) - \frac{1}{\sin (x)} - \cot (x) \cdot 1$

Step 2.1.3

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\csc (x) \sec (x) + \csc (x) - \csc (x) - \cot (x) \cdot 1$

Step 2.1.4

Multiply $- 1$ by $1$ .

$\csc (x) \sec (x) + \csc (x) - \csc (x) - \cot (x)$

Step 2.2

Subtract $\csc (x)$ from $\csc (x)$ .

$\csc (x) \sec (x) + 0 - \cot (x)$

Step 2.3

Add $\csc (x) \sec (x)$ and $0$ .

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