The value of expression `(tan alpha+sin alpha)/(2"cos"^(2)(alpha)/(2))` for `alpha=(pi)/(4)` is :

Options

A. 4
B. 3
C. 2
D. 1

Answer

To solve the expression \((\tan \alpha + \sin \alpha) / (2 \cos^2(\alpha/2))\) for \(\alpha = \frac{\pi}{4}\), we will follow these steps:

### Step 1: Substitute \(\alpha\)
Substituting \(\alpha = \frac{\pi}{4}\) into the expression, we have:
\[ \tan\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{4}\right) \]

### Step 2: Calculate \(\tan\left(\frac{\pi}{4}\right)\) and \(\sin\left(\frac{\pi}{4}\right)\)
We know that:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]
\[ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
Thus, we can substitute these values into the expression:
\[ 1 + \frac{1}{\sqrt{2}} \]

### Step 3: Calculate the Denominator
Next, we calculate the denominator \(2 \cos^2\left(\frac{\alpha}{2}\right)\):
\[ \alpha = \frac{\pi}{4} \implies \frac{\alpha}{2} = \frac{\pi}{8} \]
Using the double angle formula, we know:
\[ \cos^2\left(\frac{\pi}{8}\right) = \frac{1 + \cos\left(\frac{\pi}{4}\right)}{2} \]
Since \(\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\):
\[ \cos^2\left(\frac{\pi}{8}\right) = \frac{1 + \frac{1}{\sqrt{2}}}{2} \]
Thus:
\[ 2 \cos^2\left(\frac{\pi}{8}\right) = 1 + \frac{1}{\sqrt{2}} \]

### Step 4: Formulate the Final Expression
Now we can substitute the values back into the expression:
\[ \frac{1 + \frac{1}{\sqrt{2}}}{1 + \frac{1}{\sqrt{2}}} \]

### Step 5: Simplify the Expression
This simplifies to:
\[ 1 \]

### Conclusion
The value of the expression \((\tan \alpha + \sin \alpha) / (2 \cos^2(\alpha/2))\) for \(\alpha = \frac{\pi}{4}\) is:
\[ \boxed{1} \]