Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

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

Step 3

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

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

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Step 4.1.1.1

Multiply by $1$ .

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

Step 4.1.1.2

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

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

Step 4.1.1.3

Factor $\sin (x)$ out of $\sin (x) \cdot 1 + \sin (x) (- \cos (x) \cdot (- 1))$ .

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

Step 4.1.2

Multiply $- \cos (x) (- 1)$ .

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Step 4.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.2.2

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

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

Step 4.2

Cancel the common factor of $1 + \cos (x)$ .

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

Step 5

Rewrite $\frac{1}{\sin (x)}$ as $\csc (x)$ .

$\csc (x)$

Step 6

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{1 + \cos (- x)}{\sin (x) - \cos (x) \sin (- x)} = \csc (x)$ is an identity