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

Options

A. `pi`
B. `n pi - (pi)/(4), n in Z`
C. `(pi)/(4)`
D. `(pi)/(2)`

Answer

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

### Step 1: Rewrite the equation using trigonometric identities
We start with the given equation:
\[ 2 \sec(2\alpha) = \tan(\beta) + \cot(\beta) \]
We know that:
\[ \sec(2\alpha) = \frac{1}{\cos(2\alpha)} \]
Thus, we can rewrite the left side:
\[ 2 \sec(2\alpha) = \frac{2}{\cos(2\alpha)} \]

### Step 2: Simplify the right side using trigonometric identities
The right side can be expressed as:
\[ \tan(\beta) + \cot(\beta) = \frac{\sin(\beta)}{\cos(\beta)} + \frac{\cos(\beta)}{\sin(\beta)} \]
To combine these 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 have:
\[ \tan(\beta) + \cot(\beta) = \frac{1}{\sin(\beta) \cos(\beta)} \]

### Step 3: Set the two sides equal
Now we can equate both sides:
\[ \frac{2}{\cos(2\alpha)} = \frac{1}{\sin(\beta) \cos(\beta)} \]

### Step 4: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ 2 \sin(\beta) \cos(\beta) = \cos(2\alpha) \]

### Step 5: Use the double angle identity
We know that:
\[ \sin(2\beta) = 2 \sin(\beta) \cos(\beta) \]
Thus, we can rewrite the equation as:
\[ \sin(2\beta) = \cos(2\alpha) \]

### Step 6: Relate sine and cosine
Using the identity \( \cos(x) = \sin\left(\frac{\pi}{2} - x\right) \), we can express \( \cos(2\alpha) \) as:
\[ \cos(2\alpha) = \sin\left(\frac{\pi}{2} - 2\alpha\right) \]
Thus, we have:
\[ \sin(2\beta) = \sin\left(\frac{\pi}{2} - 2\alpha\right) \]

### Step 7: Set the angles equal
Since the sine function is equal, we can set the angles equal to each other:
\[ 2\beta = \frac{\pi}{2} - 2\alpha \]
Rearranging gives:
\[ 2\alpha + 2\beta = \frac{\pi}{2} \]

### Step 8: Solve for \( \alpha + \beta \)
Dividing the entire equation by 2:
\[ \alpha + \beta = \frac{\pi}{4} \]

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

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