Write the value of 2sin`75^(@) sin15^(@)`

Answer

To find the value of \( 2 \sin 75^\circ \sin 15^\circ \), we can use trigonometric identities. Here’s a step-by-step solution:

### Step 1: Use the Sine Angle Property
We know that:
\[ \sin(90^\circ - \theta) = \cos(\theta) \]
Thus, we can rewrite \( \sin 15^\circ \) as:
\[ \sin 15^\circ = \sin(90^\circ - 75^\circ) = \cos 75^\circ \]

### Step 2: Substitute into the Expression
Now we can substitute this back into our original expression:
\[ 2 \sin 75^\circ \sin 15^\circ = 2 \sin 75^\circ \cos 75^\circ \]

### Step 3: Apply the Double Angle Identity
We use the identity:
\[ 2 \sin \theta \cos \theta = \sin(2\theta) \]
In our case, \( \theta = 75^\circ \):
\[ 2 \sin 75^\circ \cos 75^\circ = \sin(2 \times 75^\circ) = \sin(150^\circ) \]

### Step 4: Evaluate \( \sin 150^\circ \)
We know that:
\[ \sin(180^\circ - \theta) = \sin(\theta) \]
Thus:
\[ \sin 150^\circ = \sin(180^\circ - 30^\circ) = \sin 30^\circ \]

### Step 5: Find the Value of \( \sin 30^\circ \)
The value of \( \sin 30^\circ \) is:
\[ \sin 30^\circ = \frac{1}{2} \]

### Final Result
Therefore, the value of \( 2 \sin 75^\circ \sin 15^\circ \) is:
\[ \boxed{\frac{1}{2}} \]

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