Trigonometry Examples

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

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

Step 1

Simplify the numerator.

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Step 1.1

Since

sec (- x)

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

sec (- x)

$\sec (- x)$ as

sec (x)

$\sec (x)$ .

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

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

Step 1.2

Rewrite

sec (x)

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

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

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

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

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

Step 2

Simplify the denominator.

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Step 2.1

Since

sin (- x)

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

sin (- x)

$\sin (- x)$ as

- sin (x)

$- \sin (x)$ .

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

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

Step 2.2

Since

tan (- x)

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

tan (- x)

$\tan (- x)$ as

- tan (x)

$- \tan (x)$ .

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

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

Step 2.3

Rewrite

tan (x)

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

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

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

Step 2.4

Factor

- sin (x)

$- \sin (x)$ out of

- sin (x) - \frac{sin (x)}{cos (x)}

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

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Step 2.4.1

Factor

- sin (x)

$- \sin (x)$ out of

- sin (x)

$- \sin (x)$ .

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

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

Step 2.4.2

Factor

- sin (x)

$- \sin (x)$ out of

- \frac{sin (x)}{cos (x)}

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

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

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

Step 2.4.3

Factor

- sin (x)

$- \sin (x)$ out of

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

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

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

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

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

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

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

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

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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

$\csc (x)$ .

$- \csc (x)$