If `cosectheta-cottheta=q`, then the value of `cosectheta` is

Options

A. `q+1/q`
B. `q-1/q`
C. `1/2(q+1/q)`
D. None of these

Explanation

To solve the equation \( \csc \theta - \cot \theta = q \) for \( \csc \theta \), we can follow these steps:

### Step 1: Write down the given equation
We start with the equation provided in the question:
\[ \csc \theta - \cot \theta = q \]

### Step 2: Use the trigonometric identity
We know from trigonometric identities that:
\[ \csc^2 \theta - \cot^2 \theta = 1 \]
This can be factored as:
\[ (\csc \theta - \cot \theta)(\csc \theta + \cot \theta) = 1 \]

### Step 3: Substitute the known value
From the given equation, we can substitute \( \csc \theta - \cot \theta \) with \( q \):
\[ q(\csc \theta + \cot \theta) = 1 \]

### Step 4: Solve for \( \csc \theta + \cot \theta \)
Now, we can express \( \csc \theta + \cot \theta \) in terms of \( q \):
\[ \csc \theta + \cot \theta = \frac{1}{q} \]

### Step 5: Set up a system of equations
Now we have two equations:
1. \( \csc \theta - \cot \theta = q \) (Equation 1)
2. \( \csc \theta + \cot \theta = \frac{1}{q} \) (Equation 2)

### Step 6: Add the two equations
Adding Equation 1 and Equation 2:
\[ (\csc \theta - \cot \theta) + (\csc \theta + \cot \theta) = q + \frac{1}{q} \]
This simplifies to:
\[ 2 \csc \theta = q + \frac{1}{q} \]

### Step 7: Solve for \( \csc \theta \)
Now, divide both sides by 2 to isolate \( \csc \theta \):
\[ \csc \theta = \frac{1}{2} \left( q + \frac{1}{q} \right) \]

### Final Answer
Thus, the value of \( \csc \theta \) is:
\[ \csc \theta = \frac{1}{2} \left( q + \frac{1}{q} \right) \]

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