Evaluate :
(i) `cos^(2) 25^(@) - sin^(2)65^(@) - tan^(2)45^(@)`
(ii) `((sin 77)/(cos 13))^(2) + ((cos 77)/(sin 13))^(2) - 2 cos^(2) 45^(@)`

Answer

Let's evaluate the given expressions step by step.

### Part (i): Evaluate \( \cos^2 25^\circ - \sin^2 65^\circ - \tan^2 45^\circ \)

1. **Identify the trigonometric identities**:
We know that:
\[ \sin(90^\circ - \theta) = \cos \theta \]
Therefore, we can express \( \sin^2 65^\circ \) as:
\[ \sin^2 65^\circ = \cos^2(90^\circ - 65^\circ) = \cos^2 25^\circ \]

2. **Substitute the identity into the expression**:
Now we can rewrite the expression:
\[ \cos^2 25^\circ - \sin^2 65^\circ - \tan^2 45^\circ = \cos^2 25^\circ - \cos^2 25^\circ - \tan^2 45^\circ \]

3. **Evaluate \( \tan^2 45^\circ \)**:
We know that:
\[ \tan 45^\circ = 1 \quad \Rightarrow \quad \tan^2 45^\circ = 1^2 = 1 \]

4. **Simplify the expression**:
Substituting \( \tan^2 45^\circ \) into the expression gives:
\[ \cos^2 25^\circ - \cos^2 25^\circ - 1 = 0 - 1 = -1 \]

5. **Final answer for part (i)**:
\[ \text{The answer is } -1. \]