Simplify (cos(x))/(1-tan(x))+(sin(x))/(1-cot(x))

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Trigonometry

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

Simplify each term.

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

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

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

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

To write $\frac{\cos (x)}{1 - \frac{\sin (x)}{\cos (x)}}$ as a fraction with a common denominator, multiply by $\frac{1 - \frac{\cos (x)}{\sin (x)}}{1 - \frac{\cos (x)}{\sin (x)}}$ .

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

To write $\frac{\sin (x)}{1 - \frac{\cos (x)}{\sin (x)}}$ as a fraction with a common denominator, multiply by $\frac{1 - \frac{\sin (x)}{\cos (x)}}{1 - \frac{\sin (x)}{\cos (x)}}$ .

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

Write each expression with a common denominator of $(1 - \frac{\sin (x)}{\cos (x)}) (1 - \frac{\cos (x)}{\sin (x)})$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

Combine.

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

Reorder the factors of $(1 - \frac{\sin (x)}{\cos (x)}) (1 - \frac{\cos (x)}{\sin (x)})$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Apply the distributive property.

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

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

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

Rewrite using the commutative property of multiplication.

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

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Apply the distributive property.

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

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

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

Rewrite using the commutative property of multiplication.

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

Simplify the denominator.

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Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

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