Prove that : `(4)/(3) tan ^(2) 30^(@) + sin ^(2) 60^(@) - 3 cos ^(2) 60^(@) + (3)/(4) tan^(2) 60^(@) - 2 tan^(2) 45^(@) = (25)/(36)`

Answer

To prove that

\[ \frac{4}{3} \tan^2 30^\circ + \sin^2 60^\circ - 3 \cos^2 60^\circ + \frac{3}{4} \tan^2 60^\circ - 2 \tan^2 45^\circ = \frac{25}{36} \]

we will substitute the values of the trigonometric functions for the angles involved.

### Step 1: Substitute the values of trigonometric functions

We know the following values:
- \(\tan 30^\circ = \frac{1}{\sqrt{3}}\)
- \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
- \(\cos 60^\circ = \frac{1}{2}\)
- \(\tan 60^\circ = \sqrt{3}\)
- \(\tan 45^\circ = 1\)

Now we will substitute these values into the left-hand side (LHS) of the equation.

\[ \frac{4}{3} \left(\frac{1}{\sqrt{3}}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 - 3 \left(\frac{1}{2}\right)^2 + \frac{3}{4} (\sqrt{3})^2 - 2 (1)^2 \]

### Step 2: Calculate each term

Now we will calculate each term step by step.

1. Calculate \(\tan^2 30^\circ\):
\[ \tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]
So,
\[ \frac{4}{3} \tan^2 30^\circ = \frac{4}{3} \cdot \frac{1}{3} = \frac{4}{9} \]

2. Calculate \(\sin^2 60^\circ\):
\[ \sin^2 60^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

3. Calculate \(\cos^2 60^\circ\):
\[ \cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
So,
\[ -3 \cos^2 60^\circ = -3 \cdot \frac{1}{4} = -\frac{3}{4} \]

4. Calculate \(\tan^2 60^\circ\):
\[ \tan^2 60^\circ = (\sqrt{3})^2 = 3 \]
So,
\[ \frac{3}{4} \tan^2 60^\circ = \frac{3}{4} \cdot 3 = \frac{9}{4} \]

5. Calculate \(\tan^2 45^\circ\):
\[ \tan^2 45^\circ = 1^2 = 1 \]
So,
\[ -2 \tan^2 45^\circ = -2 \cdot 1 = -2 \]

### Step 3: Combine all the terms

Now we can combine all the calculated terms:

\[ \frac{4}{9} + \frac{3}{4} - \frac{3}{4} + \frac{9}{4} - 2 \]

Notice that \(\frac{3}{4} - \frac{3}{4} = 0\), so we can simplify:

\[ \frac{4}{9} + \frac{9}{4} - 2 \]

### Step 4: Find a common denominator

The common denominator for \(9\) and \(4\) is \(36\). We will convert each term:

1. Convert \(\frac{4}{9}\):
\[ \frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36} \]

2. Convert \(\frac{9}{4}\):
\[ \frac{9}{4} = \frac{9 \times 9}{4 \times 9} = \frac{81}{36} \]

3. Convert \(-2\):
\[ -2 = -\frac{72}{36} \]

### Step 5: Combine the fractions

Now we can combine:

\[ \frac{16}{36} + \frac{81}{36} - \frac{72}{36} = \frac{16 + 81 - 72}{36} = \frac{25}{36} \]

### Conclusion

Thus, we have shown that:

\[ \frac{4}{3} \tan^2 30^\circ + \sin^2 60^\circ - 3 \cos^2 60^\circ + \frac{3}{4} \tan^2 60^\circ - 2 \tan^2 45^\circ = \frac{25}{36} \]