What is value of expression `2(sin15^(@)+sin75^(@))^(2)`?

Answer

To solve the expression \( 2(\sin 15^\circ + \sin 75^\circ)^2 \), we can follow these steps:

### Step 1: Use the sine addition formula
We know that the sum of two sine functions can be expressed using the formula:
\[ \sin a + \sin b = 2 \sin\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \]
For our case, let \( a = 15^\circ \) and \( b = 75^\circ \).

### Step 2: Calculate \( \sin 15^\circ + \sin 75^\circ \)
Using the formula:
\[ \sin 15^\circ + \sin 75^\circ = 2 \sin\left(\frac{15^\circ + 75^\circ}{2}\right) \cos\left(\frac{15^\circ - 75^\circ}{2}\right) \]
Calculating the angles:
\[ \sin 15^\circ + \sin 75^\circ = 2 \sin\left(\frac{90^\circ}{2}\right) \cos\left(\frac{-60^\circ}{2}\right) = 2 \sin(45^\circ) \cos(-30^\circ) \]

### Step 3: Substitute known values
We know that:
\[ \sin(45^\circ) = \frac{\sqrt{2}}{2}, \quad \cos(-30^\circ) = \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
So, substituting these values:
\[ \sin 15^\circ + \sin 75^\circ = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2} \]

### Step 4: Square the result
Now we need to square the result:
\[ (\sin 15^\circ + \sin 75^\circ)^2 = \left(\frac{\sqrt{6}}{2}\right)^2 = \frac{6}{4} = \frac{3}{2} \]

### Step 5: Multiply by 2
Now, we multiply by 2 as per the original expression:
\[ 2(\sin 15^\circ + \sin 75^\circ)^2 = 2 \cdot \frac{3}{2} = 3 \]

### Final Answer
Thus, the value of the expression \( 2(\sin 15^\circ + \sin 75^\circ)^2 \) is:
\[ \boxed{3} \]