To simplify the expression [tex]\(\frac{\csc(t)}{\sec(t)}\)[/tex] to a single trigonometric function, let's proceed step-by-step.
1. Recall the definitions of the trigonometric functions involved:
- [tex]\(\csc(t)\)[/tex] is the cosecant of [tex]\(t\)[/tex], which is defined as [tex]\(\csc(t) = \frac{1}{\sin(t)}\)[/tex]
- [tex]\(\sec(t)\)[/tex] is the secant of [tex]\(t\)[/tex], which is defined as [tex]\(\sec(t) = \frac{1}{\cos(t)}\)[/tex]
2. Rewrite the expression using these definitions:
[tex]\[
\frac{\csc(t)}{\sec(t)} = \frac{\frac{1}{\sin(t)}}{\frac{1}{\cos(t)}}
\][/tex]
3. Simplify the complex fraction:
To simplify the division of two fractions, we multiply by the reciprocal of the denominator:
[tex]\[
\frac{\frac{1}{\sin(t)}}{\frac{1}{\cos(t)}} = \frac{1}{\sin(t)} \times \frac{\cos(t)}{1} = \frac{\cos(t)}{\sin(t)}
\][/tex]
4. Identify the resulting trigonometric function:
The quotient of [tex]\(\cos(t)\)[/tex] and [tex]\(\sin(t)\)[/tex] is defined as the cotangent of [tex]\(t\)[/tex]:
[tex]\[
\frac{\cos(t)}{\sin(t)} = \cot(t)
\][/tex]
Therefore, the simplified expression [tex]\(\frac{\csc(t)}{\sec(t)}\)[/tex] is [tex]\(\cot(t)\)[/tex].
