`DeltaABC` is a right angled triangle, where `angle ABC = 90^@`,If `AC = 2sqrt5 and AB - BC =2`, then the value of `cos^2 A - sin^2C` is

Options

A. `1/sqrt5`
B. `sqrt5`
C. `1/2`
D. `3/5`

Answer

To solve the problem step by step, we will follow the information given in the question and apply the Pythagorean theorem to find the required value.

### Step 1: Define the sides of the triangle
Given that triangle ABC is a right-angled triangle with angle ABC = 90°, we can denote:
- \( AC = 2\sqrt{5} \) (hypotenuse)
- Let \( BC = x \) (one leg)
- Then \( AB = x + 2 \) (the other leg, since \( AB - BC = 2 \))

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

### Step 3: Expand and simplify the equation
Expanding \( (x + 2)^2 \):
\[ 20 = x^2 + 4x + 4 + x^2 \]
Combining like terms:
\[ 20 = 2x^2 + 4x + 4 \]
Rearranging the equation:
\[ 2x^2 + 4x + 4 - 20 = 0 \]
\[ 2x^2 + 4x - 16 = 0 \]
Dividing the entire equation by 2:
\[ x^2 + 2x - 8 = 0 \]

### Step 4: Factor the quadratic equation
Factoring the quadratic:
\[ (x + 4)(x - 2) = 0 \]
Setting each factor to zero gives:
\[ x + 4 = 0 \quad \text{or} \quad x - 2 = 0 \]
Thus, \( x = -4 \) (not valid for triangle sides) or \( x = 2 \).

### Step 5: Find the lengths of the sides
Since \( x = 2 \):
- \( BC = 2 \)
- \( AB = x + 2 = 2 + 2 = 4 \)

### Step 6: Calculate \( \cos^2 A \) and \( \sin^2 C \)
Using the definitions of cosine and sine:
- \( \cos A = \frac{AB}{AC} = \frac{4}{2\sqrt{5}} = \frac{2}{\sqrt{5}} \)
- \( \sin C = \frac{BC}{AC} = \frac{2}{2\sqrt{5}} = \frac{1}{\sqrt{5}} \)

Now, squaring both:
\[ \cos^2 A = \left(\frac{2}{\sqrt{5}}\right)^2 = \frac{4}{5} \]
\[ \sin^2 C = \left(\frac{1}{\sqrt{5}}\right)^2 = \frac{1}{5} \]

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

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