Rewrite in terms of sine and cosine, and simplify the expression: $\frac { \cot \beta - \cot \beta \cos \beta - \cos \beta \sin \beta } { \cos \beta \cot \beta }$
A) $\cot ^ { 2 } \beta - \cos ^ { 2 } \beta$
B) $\cos \beta - \cot \beta$
C) $\cos \beta - 1$
D) $\sin \beta - \cos \beta$
E) $\frac { \cos \beta \cot \beta } { \cos \beta + 1 }$

Question

Quizplus · Accepted Answer

We start by rewriting the expression in terms of sine and cosine:
\begin{align*}
\frac { \cot \beta - \cot \beta \cos \beta - \cos \beta \sin \beta } { \cos \beta \cot \beta } &= \frac {\frac{\cos\beta}{\sin\beta} - \frac{\cos\beta}{\sin\beta}\cos\beta - \cos\beta\sin\beta}{\frac{\cos\beta}{\sin\beta}\cdot\frac{1}{\sin\beta}}\
&= \frac{1-\cos^2\beta-\sin^2\beta}{\cos\beta\sin\beta}\
&=\frac{1-\cos^2\beta}{\cos\beta\sin\beta}\
&=\frac{\sin^2\beta}{\cos\beta\sin\beta}\
&=\frac{\sin\beta}{\cos\beta}\
&=\cos\beta-1
\end{align*}

Therefore, the answer is choice C.

Rewrite in Terms of Sine and Cosine, and Simplify the Expression