If `"2sec" (2alpha) = "tan" beta + "cot"beta`, then one of the value of `alpha + beta` is

Options

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

Answer

To solve the equation \( 2 \sec(2\alpha) = \tan(\beta) + \cot(\beta) \), we can follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ 2 \sec(2\alpha) = \tan(\beta) + \cot(\beta) \]

### Step 2: Express secant in terms of cosine
Recall that \( \sec(x) = \frac{1}{\cos(x)} \). Therefore, we can rewrite the left side:
\[ 2 \sec(2\alpha) = \frac{2}{\cos(2\alpha)} \]

### Step 3: Rewrite tangent and cotangent
Using the definitions of tangent and cotangent:
\[ \tan(\beta) = \frac{\sin(\beta)}{\cos(\beta)} \quad \text{and} \quad \cot(\beta) = \frac{\cos(\beta)}{\sin(\beta)} \]
Thus, we can express the right side as:
\[ \tan(\beta) + \cot(\beta) = \frac{\sin(\beta)}{\cos(\beta)} + \frac{\cos(\beta)}{\sin(\beta)} \]

### Step 4: Combine the right side into a single fraction
To combine the fractions, we find a common denominator:
\[ \tan(\beta) + \cot(\beta) = \frac{\sin^2(\beta) + \cos^2(\beta)}{\sin(\beta) \cos(\beta)} \]
Using the Pythagorean identity \( \sin^2(\beta) + \cos^2(\beta) = 1 \), we simplify this to:
\[ \tan(\beta) + \cot(\beta) = \frac{1}{\sin(\beta) \cos(\beta)} \]

### Step 5: Set the left side equal to the right side
Now we have:
\[ \frac{2}{\cos(2\alpha)} = \frac{1}{\sin(\beta) \cos(\beta)} \]

### Step 6: Cross-multiply
Cross-multiplying gives:
\[ 2 \sin(\beta) \cos(\beta) = \cos(2\alpha) \]

### Step 7: Use the double angle identity
Recall that \( \sin(2\beta) = 2 \sin(\beta) \cos(\beta) \). Thus, we can rewrite the equation as:
\[ \cos(2\alpha) = \sin(2\beta) \]

### Step 8: Use the complementary angle identity
We know that \( \cos(x) = \sin\left(\frac{\pi}{2} - x\right) \). Therefore, we can write:
\[ \cos(2\alpha) = \sin\left(\frac{\pi}{2} - 2\alpha\right) \]
This implies:
\[ 2\alpha = \frac{\pi}{2} - 2\beta \]

### Step 9: Rearrange the equation
Rearranging gives us:
\[ 2\alpha + 2\beta = \frac{\pi}{2} \]

### Step 10: Divide by 2
Dividing the entire equation by 2 results in:
\[ \alpha + \beta = \frac{\pi}{4} \]

### Conclusion
Thus, one of the values of \( \alpha + \beta \) is:
\[ \alpha + \beta = \frac{\pi}{4} \]