CPM Homework Help : INT3 Problem 12-112

Express $\frac { \operatorname { cos } ( x ) } { 1 - \operatorname { tan } ( x ) } - \frac { \operatorname { sin } ( x ) } { \operatorname { cot } ( x ) - 1 }$ in terms of $\sin(x)$ and $\cos(x)$ and then simplify.

Hint:

How can you change $\tan(x)$ and $\cot(x)$ to $\sin(x)$ and $\cos(x)$ ?
What ratios using those trig functions do they equal?

Step 1:

Step 2:

Simplify to create common denominators in the denominators of the larger fractions.

Step 3:

Simplify each fraction using a Giant One.

Step 4:

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

$\frac{\cos \left(x\right)}{\frac{\cos \left(x\right)-\sin \left(x\right)}{\cos \left(x\right)}}-\frac{\sin \left(x\right)}{\frac{\cos \left(x\right)-\sin \left(x\right)}{\sin \left(x\right)}}$

$\frac{\cos \left(x\right)}{\cos \left(x\right)}\cdot \frac{\cos \left(x\right)}{\frac{\cos \left(x\right)-\sin \left(x\right)}{\cos \left(x\right)}}-\frac{\sin \left(x\right)}{\frac{\cos \left(x\right)-\sin \left(x\right)}{\sin \left(x\right)}}\cdot \frac{\sin \left(x\right)}{\sin \left(x\right)}$

$\frac{\cos ^2\left(x\right)}{\cos \left(x\right)-\sin \left(x\right)}-\frac{\sin ^2\left(x\right)}{\cos \left(x\right)-\sin \left(x\right)}$

Notice you have created a common denominator.
Combine the two fractions. The numerator is a difference of two squares.
Keep simplifying.