If `cosectheta+cot theta=7`, then find `tantheta`.

Answer

To solve the equation \( \csc \theta + \cot \theta = 7 \) and find \( \tan \theta \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know that:
\[ \csc^2 \theta - \cot^2 \theta = 1 \]
This is a fundamental identity in trigonometry.

### Step 2: Factor the Difference of Squares
We can factor the left side using the difference of squares:
\[ \csc^2 \theta - \cot^2 \theta = (\csc \theta - \cot \theta)(\csc \theta + \cot \theta) \]
From the problem, we know that \( \csc \theta + \cot \theta = 7 \). Therefore, we can substitute this into our equation:
\[ (\csc \theta - \cot \theta)(7) = 1 \]

### Step 3: Solve for \( \csc \theta - \cot \theta \)
Now, we can isolate \( \csc \theta - \cot \theta \):
\[ \csc \theta - \cot \theta = \frac{1}{7} \]

### Step 4: Set Up the System of Equations
Now we have two equations:
1. \( \csc \theta + \cot \theta = 7 \)
2. \( \csc \theta - \cot \theta = \frac{1}{7} \)

### Step 5: Add the Two Equations
Adding these two equations will help eliminate \( \cot \theta \):
\[ (\csc \theta + \cot \theta) + (\csc \theta - \cot \theta) = 7 + \frac{1}{7} \]
This simplifies to:
\[ 2 \csc \theta = 7 + \frac{1}{7} \]
To combine the right side, convert \( 7 \) into a fraction:
\[ 7 = \frac{49}{7} \]
Thus:
\[ 2 \csc \theta = \frac{49}{7} + \frac{1}{7} = \frac{50}{7} \]

### Step 6: Solve for \( \csc \theta \)
Now divide both sides by 2:
\[ \csc \theta = \frac{50}{14} = \frac{25}{7} \]

### Step 7: Find \( \cot \theta \)
Substituting \( \csc \theta \) back into one of our original equations to find \( \cot \theta \):
Using \( \csc \theta + \cot \theta = 7 \):
\[ \frac{25}{7} + \cot \theta = 7 \]
Subtract \( \frac{25}{7} \) from both sides:
\[ \cot \theta = 7 - \frac{25}{7} = \frac{49}{7} - \frac{25}{7} = \frac{24}{7} \]

### Step 8: Find \( \tan \theta \)
Now we know that:
\[ \tan \theta = \frac{1}{\cot \theta} = \frac{1}{\frac{24}{7}} = \frac{7}{24} \]

### Final Answer
Thus, the value of \( \tan \theta \) is:
\[ \tan \theta = \frac{7}{24} \]