Questions: Simplify sec(t) / csc(t) to a single trig function with no fractions.

$Simplify sec(t) / csc(t) to a single trig function with no fractions.$

Transcript text: Simplify $\frac{\sec (t)}{\csc (t)}$ to a single trig function with no fractions.

Solution

Step 1: Rewrite $\sec(t)$ and $\csc(t)$ in terms of sine and cosine

\[ \sec(t) = \frac{1}{\cos(t)}, \quad \csc(t) = \frac{1}{\sin(t)} \]

Step 2: Substitute the expressions into the original fraction

\[ \frac{\sec(t)}{\csc(t)} = \frac{\frac{1}{\cos(t)}}{\frac{1}{\sin(t)}} \]

Step 3: Simplify the complex fraction

\[ \frac{\frac{1}{\cos(t)}}{\frac{1}{\sin(t)}} = \frac{1}{\cos(t)} \cdot \frac{\sin(t)}{1} = \frac{\sin(t)}{\cos(t)} \]

Step 4: Recognize the simplified expression as a single trigonometric function

\[ \frac{\sin(t)}{\cos(t)} = \tan(t) \]

$\boxed{\tan(t)}$

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