To solve the equation \(\frac{1 + \tan(t)}{\sin(t)} = 0\), we can follow these steps:

### Step 1: Understand the Equation
The equation is given as:
\[
\frac{1 + \tan(t)}{\sin(t)} = 0
\]
For a fraction to be equal to zero, the numerator must be zero (since the denominator should not be zero).

### Step 2: Set the Numerator Equal to Zero
\[
1 + \tan(t) = 0
\]
\[
\tan(t) = -1
\]

### Step 3: Solve for \( t \)
The equation \(\tan(t) = -1\) is true when \( t \) is an odd multiple of \(\frac{3\pi}{4}\), that is:
\[
t = \frac{3\pi}{4} + n\pi, \quad n \in \mathbb{Z}
\]
where \( n \) is any integer.

### Step 4: Check the Denominator
We must ensure that the denominator \(\sin(t)\) is not zero for the values of \( t \) we found. 

\(\sin(t)\) is zero at \( t = n\pi \), where \( n \) is an integer. However, \( t = \frac{3\pi}{4} + n\pi \) does not lead to \(\sin(t) = 0\), so these values are valid solutions.

### Final Answer:
The general solution for \( t \) is:
\[
t = \frac{3\pi}{4} + n\pi, \quad n \in \mathbb{Z}
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. What are the general solutions for \(\tan(t) = 1\)?
2. How do you solve \(\frac{\tan(t) - 1}{\cos(t)} = 0\)?
3. What is the period of the tangent function?
4. How does the solution change if the equation was \(\frac{\tan(t) - 1}{\sin(t)} = 0\)?
5. Can you solve the equation \(\frac{1 + \cot(t)}{\cos(t)} = 0\)?

**Tip:** When solving trigonometric equations, always check for the conditions where the denominator might be zero to avoid invalid solutions.

Explore how to solve the trigonometric equation (1 + tan(t)) / sin(t) = 0 step by step. This detailed solution covers key concepts in trigonometry, including tangent and sine functions, along with the general solution for tan(t) = -1. Ideal for undergraduate calculus students.

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