If `cottheta-tantheta=sectheta` , then `theta` is equal to

Options

A. `2npi+(3pi)/2, n in Z`
B. `npi+(-1)^npi/6, n inZ`
C. `npi+pi/2, n in Z`
D. none of these

Answer

To solve the equation \( \cot \theta - \tan \theta = \sec \theta \), we will follow these steps:

### Step 1: Rewrite the trigonometric functions in terms of sine and cosine.
The cotangent, tangent, and secant functions can be expressed as:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \sec \theta = \frac{1}{\cos \theta} \]
Substituting these into the equation gives:
\[ \frac{\cos \theta}{\sin \theta} - \frac{\sin \theta}{\cos \theta} = \frac{1}{\cos \theta} \]

### Step 2: Combine the left-hand side.
To combine the fractions on the left-hand side, find a common denominator:
\[ \frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\cos \theta} \]

### Step 3: Cross-multiply to eliminate the fractions.
Cross-multiplying gives:
\[ (\cos^2 \theta - \sin^2 \theta) \cos \theta = \sin \theta \]

### Step 4: Rearrange the equation.
Rearranging the equation leads to:
\[ \cos^3 \theta - \sin^2 \theta \cos \theta - \sin \theta = 0 \]

### Step 5: Use the Pythagorean identity.
We can use the identity \( \sin^2 \theta = 1 - \cos^2 \theta \) to express everything in terms of \( \cos \theta \):
\[ \cos^3 \theta - (1 - \cos^2 \theta) \cos \theta - \sin \theta = 0 \]
This simplifies to:
\[ \cos^3 \theta - \cos \theta + \cos^3 \theta - \sin \theta = 0 \]

### Step 6: Factor the equation.
Now we can factor the equation:
\[ \cos \theta (\cos^2 \theta - 1) - \sin \theta = 0 \]
This gives us two cases to consider: \( \cos \theta = 0 \) or \( \cos^2 \theta - 1 = \sin \theta \).

### Step 7: Solve for \( \theta \) in each case.
1. **Case 1**: If \( \cos \theta = 0 \), then:
\[ \theta = \frac{(2n + 1) \pi}{2}, \quad n \in \mathbb{Z} \]

2. **Case 2**: If \( \sin \theta = 1 \), then:
\[ \theta = \frac{\pi}{2} + 2n\pi, \quad n \in \mathbb{Z} \]

### Final Result:
The solutions for \( \theta \) are:
\[ \theta = n\pi + \frac{5\pi}{6}, \quad n \in \mathbb{Z} \]