Question

rewrite $\frac{cos\theta - sin\theta}{cos\theta}-\frac{sin\theta+cos\theta}{sin\theta}$ over a common denominator. type your answer in terms of sine and/or cosine.
$\frac{cos\theta - sin\theta}{cos\theta}-\frac{sin\theta+cos\theta}{sin\theta}=square$ (simplify your answer)

Explanation:

Step1: Find common denominator

The common denominator of $\cos\theta$ and $\sin\theta$ is $\cos\theta\sin\theta$.

Step2: Rewrite fractions

$\frac{\cos\theta - \sin\theta}{\cos\theta}-\frac{\sin\theta+\cos\theta}{\sin\theta}=\frac{(\cos\theta - \sin\theta)\sin\theta-(\sin\theta + \cos\theta)\cos\theta}{\cos\theta\sin\theta}$

Step3: Expand numerator

Expand $(\cos\theta - \sin\theta)\sin\theta$ and $(\sin\theta + \cos\theta)\cos\theta$:
$(\cos\theta - \sin\theta)\sin\theta=\cos\theta\sin\theta-\sin^{2}\theta$
$(\sin\theta + \cos\theta)\cos\theta=\sin\theta\cos\theta+\cos^{2}\theta$
Then the numerator is $\cos\theta\sin\theta-\sin^{2}\theta-(\sin\theta\cos\theta+\cos^{2}\theta)=\cos\theta\sin\theta-\sin^{2}\theta - \sin\theta\cos\theta-\cos^{2}\theta$.

Step4: Simplify numerator

$\cos\theta\sin\theta-\sin^{2}\theta - \sin\theta\cos\theta-\cos^{2}\theta=-(\sin^{2}\theta+\cos^{2}\theta)$.
Since $\sin^{2}\theta+\cos^{2}\theta = 1$, the numerator is $- 1$.

Answer:

$-\frac{1}{\cos\theta\sin\theta}$