Trigonometry Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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 $\sec (x)$ by $1$ .

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

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.

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Cancel the common factors.

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

Step 2.1.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 2.1.4

Multiply $- 1$ by $1$ .

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

Step 2.2

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

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

Step 2.3

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

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