If `2 sec 2 theta = tan Psi + cot Psi` then one of the values of `theta + Psi =`

Options

A. `pi //2`
B. `pi//4`
C. `pi//3`
D. none

Answer

To solve the equation \( 2 \sec 2\theta = \tan \Psi + \cot \Psi \), we will follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ 2 \sec 2\theta = \tan \Psi + \cot \Psi \]
We know that:
\[ \tan \Psi = \frac{\sin \Psi}{\cos \Psi} \quad \text{and} \quad \cot \Psi = \frac{\cos \Psi}{\sin \Psi} \]
Thus, we can rewrite the right-hand side:
\[ \tan \Psi + \cot \Psi = \frac{\sin \Psi}{\cos \Psi} + \frac{\cos \Psi}{\sin \Psi} \]

### Step 2: Find a common denominator
To combine the fractions, we find a common denominator:
\[ \tan \Psi + \cot \Psi = \frac{\sin^2 \Psi + \cos^2 \Psi}{\sin \Psi \cos \Psi} \]
Using the Pythagorean identity \( \sin^2 \Psi + \cos^2 \Psi = 1 \), we simplify this to:
\[ \tan \Psi + \cot \Psi = \frac{1}{\sin \Psi \cos \Psi} \]

### Step 3: Substitute back into the equation
Now, substituting this back into the equation gives us:
\[ 2 \sec 2\theta = \frac{1}{\sin \Psi \cos \Psi} \]
Since \( \sec 2\theta = \frac{1}{\cos 2\theta} \), we can rewrite the left-hand side:
\[ \frac{2}{\cos 2\theta} = \frac{1}{\sin \Psi \cos \Psi} \]

### Step 4: Cross-multiply to eliminate fractions
Cross-multiplying gives:
\[ 2 \sin \Psi \cos \Psi = \cos 2\theta \]

### Step 5: Use the double angle identity
Using the double angle identity for sine, \( \sin 2\Psi = 2 \sin \Psi \cos \Psi \), we can rewrite the equation:
\[ \sin 2\Psi = \cos 2\theta \]

### Step 6: Relate the angles
From the equation \( \sin 2\Psi = \cos 2\theta \), we can use the co-function identity:
\[ \sin 2\Psi = \cos\left(\frac{\pi}{2} - 2\theta\right) \]
This implies:
\[ 2\Psi = \frac{\pi}{2} - 2\theta \]

### Step 7: Solve for \( \theta + \Psi \)
Rearranging gives:
\[ 2\theta + 2\Psi = \frac{\pi}{2} \]
Dividing through by 2:
\[ \theta + \Psi = \frac{\pi}{4} \]

### Final Answer
Thus, one of the values of \( \theta + \Psi \) is:
\[ \boxed{\frac{\pi}{4}} \]