Find the value of 'a' which the system of equation ` sin x * cos y=a^(2) and sin y* cos x =a ` have a solution

Answer

To find the value of 'a' for which the system of equations \( \sin x \cos y = a^2 \) and \( \sin y \cos x = a \) has a solution, we will follow these steps:

### Step 1: Write down the equations
The given equations are:
1. \( \sin x \cos y = a^2 \) (Equation 1)
2. \( \sin y \cos x = a \) (Equation 2)

### Step 2: Add the two equations
We can add both equations:
\[ \sin x \cos y + \sin y \cos x = a^2 + a \]

### Step 3: Use the sine addition formula
The left-hand side can be rewritten using the sine addition formula:
\[ \sin(x + y) = \sin x \cos y + \sin y \cos x \]
Thus, we have:
\[ \sin(x + y) = a^2 + a \]

### Step 4: Analyze the range of the sine function
The sine function has a range of \([-1, 1]\). Therefore, we can set up the inequalities:
\[ -1 \leq a^2 + a \leq 1 \]

### Step 5: Solve the inequalities
We will solve the two inequalities separately.

#### Inequality 1: \( a^2 + a \geq -1 \)
Rearranging gives:
\[ a^2 + a + 1 \geq 0 \]
The discriminant \( D \) of the quadratic \( a^2 + a + 1 \) is:
\[ D = 1^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]
Since the discriminant is negative, the quadratic \( a^2 + a + 1 \) is always positive. Thus, this inequality is satisfied for all \( a \).

#### Inequality 2: \( a^2 + a \leq 1 \)
Rearranging gives:
\[ a^2 + a - 1 \leq 0 \]
The discriminant \( D \) of the quadratic \( a^2 + a - 1 \) is:
\[ D = 1^2 - 4 \cdot 1 \cdot (-1) = 1 + 4 = 5 \]
Since the discriminant is positive, we can find the roots using the quadratic formula:
\[ a = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{5}}{2} \]
Thus, the roots are:
\[ a_1 = \frac{-1 - \sqrt{5}}{2}, \quad a_2 = \frac{-1 + \sqrt{5}}{2} \]

### Step 6: Determine the interval for 'a'
The quadratic \( a^2 + a - 1 \) opens upwards (as the coefficient of \( a^2 \) is positive). Therefore, the inequality \( a^2 + a - 1 \leq 0 \) is satisfied between the roots:
\[ \frac{-1 - \sqrt{5}}{2} \leq a \leq \frac{-1 + \sqrt{5}}{2} \]

### Conclusion
The values of \( a \) for which the system of equations has a solution are:
\[ a \in \left[ \frac{-1 - \sqrt{5}}{2}, \frac{-1 + \sqrt{5}}{2} \right] \]