We need to simplify the given expression:

$\frac{\sin(t)}{\sec(t) - \cos(t)}$

to a single trigonometric function with no fractions.

Step 1: Rewrite $\sec(t)$ using its definition

Recall that $\sec(t) = \frac{1}{\cos(t)}$. Substituting into the denominator:

$\frac{\sin(t)}{\sec(t) - \cos(t)} = \frac{\sin(t)}{\frac{1}{\cos(t)} - \cos(t)}.$

Step 2: Combine terms in the denominator

To combine terms in the denominator, find a common denominator for $\frac{1}{\cos(t)}$ and $\cos(t)$:

$\frac{1}{\cos(t)} - \cos(t) = \frac{1 - \cos^2(t)}{\cos(t)}.$

Step 3: Recognize a Pythagorean identity

From the Pythagorean identity:

$\sin^2(t) + \cos^2(t) = 1 \implies 1 - \cos^2(t) = \sin^2(t).$

Substitute this into the denominator:

$\frac{1 - \cos^2(t)}{\cos(t)} = \frac{\sin^2(t)}{\cos(t)}.$

So the expression becomes:

$\frac{\sin(t)}{\frac{\sin^2(t)}{\cos(t)}}.$

Step 4: Simplify the complex fraction

To simplify $\frac{\sin(t)}{\frac{\sin^2(t)}{\cos(t)}}$, multiply by the reciprocal of the denominator:

$\frac{\sin(t)}{\frac{\sin^2(t)}{\cos(t)}} = \sin(t) \cdot \frac{\cos(t)}{\sin^2(t)}.$

Simplify the terms:

$\sin(t) \cdot \frac{\cos(t)}{\sin^2(t)} = \frac{\sin(t) \cos(t)}{\sin^2(t)}.$

Cancel one factor of $\sin(t)$ from the numerator and denominator:

$\frac{\sin(t) \cos(t)}{\sin^2(t)} = \frac{\cos(t)}{\sin(t)}.$

Step 5: Simplify to a single trigonometric function

The expression $\frac{\cos(t)}{\sin(t)}$ is the cotangent function:

$\frac{\cos(t)}{\sin(t)} = \cot(t).$

Final Answer:

The simplified expression is:

$\cot(t).$

Do you want further details or clarification on any step?

Here are some related questions for further practice:

1. Simplify $\frac{\sec(t)}{\sec(t) + \tan(t)}$ into a single trigonometric function.
2. Prove the identity: $\frac{\sin^2(t)}{1 - \cos(t)} = 1 + \cos(t)$.
3. Simplify $\frac{\sin(t)}{\csc(t) - 1}$.
4. Verify: $\frac{\sec(t) - \cos(t)}{\sec(t)} = \sin^2(t)$.
5. Simplify $\frac{1 - \cos^2(t)}{\sin(t)}$.

Tip:

When simplifying trigonometric expressions, always check for Pythagorean identities like $1 - \cos^2(t) = \sin^2(t)$ and reciprocal definitions to simplify fractions.