If sin `theta = cos theta` , then find the value of 2 tan `theta + cos^(2) theta`.

Answer

To solve the problem where \( \sin \theta = \cos \theta \) and we need to find the value of \( 2 \tan \theta + \cos^2 \theta \), we can follow these steps:

### Step 1: Use the given equation
We start with the equation:
\[ \sin \theta = \cos \theta \]

### Step 2: Divide both sides by \( \cos \theta \)
Dividing both sides by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):
\[ \frac{\sin \theta}{\cos \theta} = 1 \]
This simplifies to:
\[ \tan \theta = 1 \]

### Step 3: Find the angle \( \theta \)
From the equation \( \tan \theta = 1 \), we know that:
\[ \theta = 45^\circ \quad \text{(or } \theta = \frac{\pi}{4} \text{ radians)} \]

### Step 4: Calculate \( \tan \theta \)
Since \( \theta = 45^\circ \):
\[ \tan 45^\circ = 1 \]

### Step 5: Calculate \( \cos^2 \theta \)
Now we calculate \( \cos^2 \theta \):
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]
Thus,
\[ \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 6: Substitute values into the expression
Now substitute \( \tan \theta \) and \( \cos^2 \theta \) into the expression \( 2 \tan \theta + \cos^2 \theta \):
\[ 2 \tan \theta + \cos^2 \theta = 2(1) + \frac{1}{2} \]
This simplifies to:
\[ 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \]

### Final Answer
Thus, the value of \( 2 \tan \theta + \cos^2 \theta \) is:
\[ \frac{5}{2} \]

---