In a right angled triangle `/_B` is right angle and `AC=2sqrt(5)` cm. If `AB-BC=2` cm then what is the value of `(cos^(2)A-cos^(2)C)` ?

Options

A. `(3)/(5)`
B. `(6)/(5)`
C. `(3)/(10)`
D. `(2)/(5)`

Answer

To solve the problem step by step, we will analyze the right-angled triangle \( \triangle ABC \) where \( \angle B \) is the right angle, \( AC = 2\sqrt{5} \) cm, and \( AB - BC = 2 \) cm. We need to find the value of \( \cos^2 A - \cos^2 C \).

### Step 1: Define the sides of the triangle
Let:
- \( AB = x \) (the side opposite angle C)
- \( BC = y \) (the side opposite angle A)

From the problem, we know:
\[ x - y = 2 \] (1)

### Step 2: Use the Pythagorean theorem
According to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the known values:
\[ (2\sqrt{5})^2 = x^2 + y^2 \]
\[ 20 = x^2 + y^2 \] (2)

### Step 3: Solve the system of equations
We have two equations:
1. \( x - y = 2 \) (1)
2. \( x^2 + y^2 = 20 \) (2)

From equation (1), we can express \( x \) in terms of \( y \):
\[ x = y + 2 \]

Substituting this into equation (2):
\[ (y + 2)^2 + y^2 = 20 \]
Expanding:
\[ y^2 + 4y + 4 + y^2 = 20 \]
Combining like terms:
\[ 2y^2 + 4y + 4 = 20 \]
Subtracting 20 from both sides:
\[ 2y^2 + 4y - 16 = 0 \]
Dividing the entire equation by 2:
\[ y^2 + 2y - 8 = 0 \]

### Step 4: Factor the quadratic equation
We can factor the quadratic:
\[ (y + 4)(y - 2) = 0 \]
Thus, \( y = -4 \) or \( y = 2 \). Since lengths cannot be negative, we have:
\[ y = 2 \]

### Step 5: Find \( x \)
Using \( y = 2 \) in equation (1):
\[ x - 2 = 2 \]
Thus, \( x = 4 \).

### Step 6: Calculate \( \cos^2 A \) and \( \cos^2 C \)
Now we can find \( \cos^2 A \) and \( \cos^2 C \):
- \( \cos A = \frac{BC}{AC} = \frac{y}{AC} = \frac{2}{2\sqrt{5}} = \frac{1}{\sqrt{5}} \)
- \( \cos^2 A = \left(\frac{1}{\sqrt{5}}\right)^2 = \frac{1}{5} \)

- \( \cos C = \frac{AB}{AC} = \frac{x}{AC} = \frac{4}{2\sqrt{5}} = \frac{2}{\sqrt{5}} \)
- \( \cos^2 C = \left(\frac{2}{\sqrt{5}}\right)^2 = \frac{4}{5} \)

### Step 7: Calculate \( \cos^2 A - \cos^2 C \)
Now we compute:
\[ \cos^2 A - \cos^2 C = \frac{1}{5} - \frac{4}{5} = -\frac{3}{5} \]

### Final Answer
Thus, the value of \( \cos^2 A - \cos^2 C \) is:
\[ \boxed{-\frac{3}{5}} \]