Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Factor $- \sin (x)$ out of $- \sin (x) (1) - \sin (x) (\frac{1}{\cos (x)})$ .

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

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

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $1$ and $- 1$ .

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Step 3.2.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 4

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

$- \csc (x)$