Precalculus Examples

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

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

Step 1

Start on the left side.

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

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

Step 2

Since

sin (- x)

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

sin (- x)

$\sin (- x)$ as

- sin (x)

$- \sin (x)$ .

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

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

Step 3

Write

tan (x)

$\tan (x)$ in sines and cosines using the quotient identity.

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

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Multiply the numerator and denominator of the fraction by

cos (x)

$\cos (x)$ .

Tap for more steps...

Step 4.1.1

Multiply

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

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

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

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

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

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

Step 4.1.2

Combine.

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

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

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

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

Step 4.2

Apply the distributive property.

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

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

Step 4.3

Cancel the common factor of

cos (x)

$\cos (x)$ .

Tap for more steps...

Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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

Step 4.4

Simplify the numerator.

Tap for more steps...

Step 4.4.1

Factor

$\sin (x)$ out of

$\sin (x) + \cos (x) (- - \sin (x))$ .

Tap for more steps...

Step 4.4.1.1

Multiply by

$1$ .

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

Step 4.4.1.2

Factor

$\sin (x)$ out of

$\cos (x) (- - \sin (x))$ .

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

Step 4.4.1.3

Factor

$\sin (x)$ out of

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

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

Step 4.4.2

Multiply

$- 1$ by

$- 1$ .

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

Step 4.4.3

Multiply

$\cos (x)$ by

$1$ .

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

Step 4.5

Factor

$\cos (x)$ out of

$\cos (x) \cdot 1 + \cos (x) \cos (x)$ .

Tap for more steps...

Step 4.5.1

Factor

$\cos (x)$ out of

$\cos (x) \cdot 1$ .

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

Step 4.5.2

Factor

$\cos (x)$ out of

$\cos (x) \cos (x)$ .

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

Step 4.5.3

Factor

$\cos (x)$ out of

$\cos (x) (1) + \cos (x) (\cos (x))$ .

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

Step 4.6

Cancel the common factor of

$1 + \cos (x)$ .

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

Step 5

Rewrite

$\frac{\sin (x)}{\cos (x)}$ as

$\tan (x)$ .

$\tan (x)$

Step 6

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

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