Solve for \( x \) in the equation \( \frac{\cos(x)}{\cos(x) - \sin(x)} = 2 \)

Step 1 ：Step 1: Rearrange the equation to have 0 on one side: \( \frac{\cos(x)}{\cos(x) - \sin(x)} - 2 = 0 \)

Step 2 ：Step 2: Combine the two fractions on the left side of the equation: \( \frac{2\cos(x) - 2\sin(x) - \cos(x)}{\cos(x) - \sin(x)} = 0 \)

Step 3 ：Step 3: Simplify the numerator: \( \frac{\cos(x) - 2\sin(x)}{\cos(x) - \sin(x)} = 0 \)

Step 4 ：Step 4: Use partial fraction decomposition to split the fraction: \( \frac{1 - 2\tan(x)}{1 - \tan(x)} = 0 \)

Step 5 ：Step 5: Simplify the equation: \( 1 - 2\tan(x) = 0 \)

Step 6 ：Step 6: Solve for \( \tan(x) \): \( \tan(x) = \frac{1}{2} \)

Step 7 ：Step 7: Solve for \( x \): \( x = \arctan\left(\frac{1}{2}\right) \)