If `2sec 2theta=tan phi+cotphi`, then one of the values of `theta+phi` is

Options

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

Explanation

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

### Step 1: Rewrite the trigonometric functions
We start by rewriting the secant, tangent, and cotangent in terms of sine and cosine:
\[ 2 \sec 2\theta = \frac{2}{\cos 2\theta} \]
\[ \tan \phi = \frac{\sin \phi}{\cos \phi}, \quad \cot \phi = \frac{\cos \phi}{\sin \phi} \]
Thus, we can rewrite the equation as:
\[ \frac{2}{\cos 2\theta} = \frac{\sin \phi}{\cos \phi} + \frac{\cos \phi}{\sin \phi} \]

### Step 2: Combine the right-hand side
To combine the right-hand side, we find a common denominator:
\[ \frac{\sin \phi}{\cos \phi} + \frac{\cos \phi}{\sin \phi} = \frac{\sin^2 \phi + \cos^2 \phi}{\sin \phi \cos \phi} \]
Using the Pythagorean identity \( \sin^2 \phi + \cos^2 \phi = 1 \), we simplify this to:
\[ \frac{1}{\sin \phi \cos \phi} \]

### Step 3: Set the two sides equal
Now we have:
\[ \frac{2}{\cos 2\theta} = \frac{1}{\sin \phi \cos \phi} \]

### Step 4: Cross-multiply
Cross-multiplying gives us:
\[ 2 \sin \phi \cos \phi = \cos 2\theta \]

### Step 5: Use the double angle identity
Recall the double angle identity for sine:
\[ \sin 2\phi = 2 \sin \phi \cos \phi \]
Thus, we can rewrite the equation as:
\[ \sin 2\phi = \cos 2\theta \]

### Step 6: Relate angles
From the equation \( \sin 2\phi = \cos 2\theta \), we can use the co-function identity:
\[ \sin 2\phi = \sin\left(\frac{\pi}{2} - 2\theta\right) \]
This implies:
\[ 2\phi = \frac{\pi}{2} - 2\theta \]
Rearranging gives:
\[ 2\theta + 2\phi = \frac{\pi}{2} \]
Dividing through by 2 gives:
\[ \theta + \phi = \frac{\pi}{4} \]

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

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