To simplify the given expression [tex]\(\frac{\csc (\theta)}{\cot (\theta)}\)[/tex], let's start by recalling the definitions and relationships of the trigonometric functions involved:
1. [tex]\(\csc (\theta)\)[/tex] is the cosecant function, which is the reciprocal of the sine function. Therefore, we have:
[tex]\[
\csc (\theta) = \frac{1}{\sin (\theta)}
\][/tex]
2. [tex]\(\cot (\theta)\)[/tex] is the cotangent function, which is the ratio of the cosine function to the sine function. Therefore, we have:
[tex]\[
\cot (\theta) = \frac{\cos (\theta)}{\sin (\theta)}
\][/tex]
Now we substitute these definitions into the original expression:
[tex]\[
\frac{\csc (\theta)}{\cot (\theta)} = \frac{\frac{1}{\sin (\theta)}}{\frac{\cos (\theta)}{\sin (\theta)}}
\][/tex]
We can simplify this complex fraction by multiplying the numerator by the reciprocal of the denominator. So we have:
[tex]\[
\frac{\csc (\theta)}{\cot (\theta)} = \frac{1}{\sin (\theta)} \times \frac{\sin (\theta)}{\cos (\theta)}
\][/tex]
Here, the [tex]\(\sin (\theta)\)[/tex] terms in the numerator and the denominator cancel out each other:
[tex]\[
\frac{1}{\cos (\theta)}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\frac{\csc (\theta)}{\cot (\theta)} = \frac{1}{\cos (\theta)}
\][/tex]
