Find the value of sin `15^@+ cos 15^@`

Answer

To find the value of \( \sin 15^\circ + \cos 15^\circ \), we can use the sine addition formula. Here’s a step-by-step solution:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sin 15^\circ + \cos 15^\circ \]
We can express this as:
\[ \sin 15^\circ + \cos 15^\circ = \sin 15^\circ + \sin 90^\circ \cdot \cos 15^\circ \]

### Step 2: Use the sine addition formula
We can use the sine addition formula:
\[ \sin A + \sin B = \sin(A + B) \]
In our case, we can set \( A = 15^\circ \) and \( B = 90^\circ \):
\[ \sin 15^\circ + \cos 15^\circ = \sin(15^\circ + 45^\circ) \cdot \sqrt{2} \]
This is because \( \sin A + \cos A = \sqrt{2} \sin(A + 45^\circ) \).

### Step 3: Calculate the angle
Now we have:
\[ \sin(15^\circ + 45^\circ) = \sin 60^\circ \]
We know that:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]

### Step 4: Multiply by \(\sqrt{2}\)
Now substituting back, we have:
\[ \sin 15^\circ + \cos 15^\circ = \sqrt{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2} \]

### Final Result
Thus, the value of \( \sin 15^\circ + \cos 15^\circ \) is:
\[ \frac{\sqrt{6}}{2} \]