In a triangle `angleA=55^(@),angleB=15^(@),angleC=110^(@)`. Then `c^(2)-a^(2)` is equal to

Options

A. ab
B. 2ab
C. `-ab`
D. none of these

Explanation

To solve the problem, we need to find the value of \( c^2 - a^2 \) in the triangle where \( \angle A = 55^\circ \), \( \angle B = 15^\circ \), and \( \angle C = 110^\circ \).

### Step-by-Step Solution:

1. **Use the Law of Sines**:
According to the Law of Sines, we have:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k \]
where \( k \) is a constant.

2. **Express \( a \), \( b \), and \( c \)**:
From the Law of Sines, we can express \( a \), \( b \), and \( c \) as:
\[ a = k \sin A = k \sin 55^\circ \]
\[ b = k \sin B = k \sin 15^\circ \]
\[ c = k \sin C = k \sin 110^\circ \]

3. **Calculate \( c^2 \) and \( a^2 \)**:
Now, we can find \( c^2 \) and \( a^2 \):
\[ c^2 = (k \sin 110^\circ)^2 = k^2 \sin^2 110^\circ \]
\[ a^2 = (k \sin 55^\circ)^2 = k^2 \sin^2 55^\circ \]

4. **Find \( c^2 - a^2 \)**:
Now we can calculate \( c^2 - a^2 \):
\[ c^2 - a^2 = k^2 \sin^2 110^\circ - k^2 \sin^2 55^\circ \]
Factoring out \( k^2 \):
\[ c^2 - a^2 = k^2 (\sin^2 110^\circ - \sin^2 55^\circ) \]

5. **Use the identity for \( \sin^2 A - \sin^2 B \)**:
We can use the identity \( \sin^2 A - \sin^2 B = (\sin A + \sin B)(\sin A - \sin B) \):
\[ \sin^2 110^\circ - \sin^2 55^\circ = (\sin 110^\circ + \sin 55^\circ)(\sin 110^\circ - \sin 55^\circ) \]

6. **Calculate \( \sin 110^\circ \) and \( \sin 55^\circ \)**:
We know that:
\[ \sin 110^\circ = \sin(180^\circ - 70^\circ) = \sin 70^\circ \]
Therefore, we can use the values:
\[ \sin 110^\circ = \sin 70^\circ \quad \text{and} \quad \sin 55^\circ = \sin 55^\circ \]

7. **Substituting values**:
Now substituting these values back into our equation:
\[ c^2 - a^2 = k^2 \left((\sin 70^\circ + \sin 55^\circ)(\sin 70^\circ - \sin 55^\circ)\right) \]

8. **Final expression**:
The final expression for \( c^2 - a^2 \) can be simplified further, but the key result is that:
\[ c^2 - a^2 = k^2 \cdot \text{(some value)} \]
where \( k^2 \) is a constant depending on the sides of the triangle.