Simplify  
1
+
tan
(
t
)
1
+
cot
(
t
)
  to a single trig function.

To simplify the expression

\[
1 + \frac{\tan(t)}{1 + \cot(t)},
\]

we can follow these steps:

### Step 1: Express everything in terms of sine and cosine.
- Recall that \(\tan(t) = \frac{\sin(t)}{\cos(t)}\) and \(\cot(t) = \frac{\cos(t)}{\sin(t)}\).
  
So, the given expression becomes:

\[
1 + \frac{\frac{\sin(t)}{\cos(t)}}{1 + \frac{\cos(t)}{\sin(t)}}.
\]

### Step 2: Simplify the denominator of the fraction.
The denominator \(1 + \frac{\cos(t)}{\sin(t)}\) can be rewritten as:

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

Thus, the entire expression becomes:

\[
1 + \frac{\frac{\sin(t)}{\cos(t)}}{\frac{\sin(t) + \cos(t)}{\sin(t)}}.
\]

### Step 3: Simplify the complex fraction.
Now, simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator:

\[
1 + \frac{\frac{\sin(t)}{\cos(t)} \cdot \sin(t)}{\sin(t) + \cos(t)} = 1 + \frac{\sin^2(t)}{\cos(t)(\sin(t) + \cos(t))}.
\]

### Step 4: Combine terms.
Now combine the \(1\) (as \(\frac{\sin(t) + \cos(t)}{\sin(t) + \cos(t)}\)) with the fraction:

\[
\frac{(\sin(t) + \cos(t))\cos(t) + \sin^2(t)}{\cos(t)(\sin(t) + \cos(t))}.
\]

### Step 5: Simplify the numerator.
The numerator becomes:

\[
\sin(t)\cos(t) + \cos^2(t) + \sin^2(t).
\]

Using the Pythagorean identity \(\sin^2(t) + \cos^2(t) = 1\), this simplifies to:

\[
\sin(t)\cos(t) + 1.
\]

Thus, the entire expression becomes:

\[
\frac{\sin(t)\cos(t) + 1}{\cos(t)(\sin(t) + \cos(t))}.
\]

### Final Simplified Expression:

This expression can't be simplified further into a single basic trigonometric function, but it is the most simplified form of the given expression.

Would you like more details, or do you have any questions?

### Related Questions:
1. What are the steps to simplify other trigonometric identities?
2. How can I simplify an expression involving both \(\sin\) and \(\cos\)?
3. What is the Pythagorean identity and how is it used?
4. How do you handle complex fractions in trigonometric expressions?
5. What are some key trigonometric identities to remember?

### Tip:
Always express trigonometric functions in terms of sine and cosine when simplifying—this often makes the process clearer!

Simplify 1 + tan(t) / (1 + cot(t)) to a single trig function.

Simplify the trigonometric expression 1 + tan(t) / (1 + cot(t)) step-by-step using basic trigonometric identities. Learn how to express the solution in terms of sine and cosine, and apply the Pythagorean identity for simplification. Suitable for Grades 10-12 students.

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