The value of `sin^2 12^0+sin^2 21^0+sin^2 39^0+sin^2 48^0-sin^2 9^0-sin^2 18^0` is _______

Explanation

To solve the expression \( \sin^2 12^\circ + \sin^2 21^\circ + \sin^2 39^\circ + \sin^2 48^\circ - \sin^2 9^\circ - \sin^2 18^\circ \), we can use some trigonometric identities and properties. Here's a step-by-step breakdown of the solution:

### Step 1: Rewrite the Sine Squares
We can use the identity \( \sin^2 x = 1 - \cos^2 x \) to rewrite the sine squares:
\[ \sin^2 12^\circ = 1 - \cos^2 12^\circ \]
\[ \sin^2 21^\circ = 1 - \cos^2 21^\circ \]
\[ \sin^2 39^\circ = 1 - \cos^2 39^\circ \]
\[ \sin^2 48^\circ = 1 - \cos^2 48^\circ \]
\[ \sin^2 9^\circ = 1 - \cos^2 9^\circ \]
\[ \sin^2 18^\circ = 1 - \cos^2 18^\circ \]

### Step 2: Substitute into the Expression
Substituting these into the original expression gives:
\[ (1 - \cos^2 12^\circ) + (1 - \cos^2 21^\circ) + (1 - \cos^2 39^\circ) + (1 - \cos^2 48^\circ) - (1 - \cos^2 9^\circ) - (1 - \cos^2 18^\circ) \]

### Step 3: Simplify the Expression
Combining the constants and simplifying:
\[ 4 - (\cos^2 12^\circ + \cos^2 21^\circ + \cos^2 39^\circ + \cos^2 48^\circ) + (\cos^2 9^\circ + \cos^2 18^\circ) \]

### Step 4: Grouping Terms
This simplifies to:
\[ 4 - (\cos^2 12^\circ + \cos^2 21^\circ + \cos^2 39^\circ + \cos^2 48^\circ - \cos^2 9^\circ - \cos^2 18^\circ) \]

### Step 5: Use Trigonometric Identities
Now, we can use the identity \( \cos^2 x = \frac{1 + \cos 2x}{2} \) to express the cosines in terms of their double angles:
\[ \cos^2 12^\circ = \frac{1 + \cos 24^\circ}{2}, \quad \cos^2 21^\circ = \frac{1 + \cos 42^\circ}{2}, \quad \cos^2 39^\circ = \frac{1 + \cos 78^\circ}{2}, \quad \cos^2 48^\circ = \frac{1 + \cos 96^\circ}{2} \]
\[ \cos^2 9^\circ = \frac{1 + \cos 18^\circ}{2}, \quad \cos^2 18^\circ = \frac{1 + \cos 36^\circ}{2} \]

### Step 6: Substitute Back and Simplify
Substituting these values back into the expression will yield a more manageable form. However, for simplicity, we can evaluate the original sine squares directly using a calculator or trigonometric tables.

### Final Calculation
After evaluating the sine squares:
\[ \sin^2 12^\circ + \sin^2 21^\circ + \sin^2 39^\circ + \sin^2 48^\circ - \sin^2 9^\circ - \sin^2 18^\circ \approx 1 \]

Thus, the final answer is:
\[ \boxed{1} \]