Find the value of : `3 sin^(2) 20^(@) - 2 tan^(2) 45^(@) + 3 sin^(2) 70^(@)` .

Answer

To solve the expression \(3 \sin^2 20^\circ - 2 \tan^2 45^\circ + 3 \sin^2 70^\circ\), we can follow these steps:

### Step 1: Evaluate \(\tan^2 45^\circ\)
We know that \(\tan 45^\circ = 1\). Therefore:
\[ \tan^2 45^\circ = 1^2 = 1 \]

### Step 2: Substitute \(\tan^2 45^\circ\) into the expression
Now, substituting this value into the expression, we get:
\[ 3 \sin^2 20^\circ - 2(1) + 3 \sin^2 70^\circ \]
This simplifies to:
\[ 3 \sin^2 20^\circ - 2 + 3 \sin^2 70^\circ \]

### Step 3: Use the identity \(\sin(90^\circ - \theta) = \cos(\theta)\)
We know that \(\sin 70^\circ = \cos 20^\circ\). Therefore:
\[ \sin^2 70^\circ = \cos^2 20^\circ \]

### Step 4: Substitute \(\sin^2 70^\circ\) into the expression
Now substituting \(\sin^2 70^\circ\) into the expression gives:
\[ 3 \sin^2 20^\circ - 2 + 3 \cos^2 20^\circ \]

### Step 5: Combine \(\sin^2\) and \(\cos^2\)
Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can rewrite the expression:
\[ 3(\sin^2 20^\circ + \cos^2 20^\circ) - 2 \]
This simplifies to:
\[ 3(1) - 2 = 3 - 2 \]

### Step 6: Final Calculation
Thus, we find:
\[ 3 - 2 = 1 \]

### Final Answer
The value of the expression is:
\[ \boxed{1} \]