Trigonometry Examples

$(1 - \sin (x)) (1 + \csc (x))$

Step 1

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

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

Step 2

Expand $(1 - \sin (x)) (1 + \frac{1}{\sin (x)})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $1$ by $1$ .

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

Step 3.1.2

Multiply $\frac{1}{\sin (x)}$ by $1$ .

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

Step 3.1.3

Multiply $- 1$ by $1$ .

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

Step 3.1.4

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

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Step 3.1.4.1

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

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

Step 3.1.4.2

Cancel the common factor.

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

Step 3.1.4.3

Rewrite the expression.

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

Step 3.2

Subtract $1$ from $1$ .

$0 + \frac{1}{\sin (x)} - \sin (x)$

Step 3.3

Add $0$ and $\frac{1}{\sin (x)}$ .

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

Step 4

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

$\csc (x) - \sin (x)$