Consider the equation `4^(sinx) + 3^(sec y) = 11 and 5(16)^(sin x) - 2(3)^(sec y) = 2`.
Then, the exhaustive values of x and y respectively are :

Options

A. `x = n pi + pi/6 , y = 2mx + pi/3, m , n inz`
B. `x = npi - pi/6, y = 2mx + pi/3, m,n inz`
C. `x = n pi + (-1)^npi/6, y = 2 mxpm pi/3' m,ninz`
D. `x = npi + (-1)^n pi/6, y = 2mx - pi/3, m, n in z`

Answer

To solve the equations given in the problem, we will follow a systematic approach. The equations are:

1. \( 4^{\sin x} + 3^{\sec y} = 11 \)
2. \( 5(16)^{\sin x} - 2(3)^{\sec y} = 2 \)

### Step 1: Define Variables
Let:
- \( m = 4^{\sin x} \)
- \( n = 3^{\sec y} \)

Then, the first equation can be rewritten as:
\[ m + n = 11 \]

### Step 2: Rewrite the Second Equation
The second equation can be rewritten using the definitions of \( m \) and \( n \):
\[ 5(16)^{\sin x} - 2(3)^{\sec y} = 2 \]
Since \( 16^{\sin x} = (4^2)^{\sin x} = (4^{\sin x})^2 = m^2 \), we can rewrite the equation as:
\[ 5(4^{\sin x})^2 - 2(3^{\sec y}) = 2 \]
Thus, substituting \( m \) and \( n \):
\[ 5m^2 - 2n = 2 \]

### Step 3: Solve the System of Equations
Now we have a system of equations:
1. \( m + n = 11 \)
2. \( 5m^2 - 2n = 2 \)

From the first equation, we can express \( n \) in terms of \( m \):
\[ n = 11 - m \]

Substituting \( n \) into the second equation:
\[ 5m^2 - 2(11 - m) = 2 \]
Expanding this gives:
\[ 5m^2 - 22 + 2m = 2 \]
Rearranging this leads to:
\[ 5m^2 + 2m - 24 = 0 \]

### Step 4: Factor the Quadratic Equation
Now we will factor the quadratic equation:
\[ 5m^2 + 12m - 10m - 24 = 0 \]
This can be factored as:
\[ (5m + 12)(m - 2) = 0 \]

### Step 5: Find the Values of \( m \)
Setting each factor to zero gives:
1. \( 5m + 12 = 0 \) → \( m = -\frac{12}{5} \) (not valid since \( m \) must be positive)
2. \( m - 2 = 0 \) → \( m = 2 \)

### Step 6: Find the Corresponding Value of \( n \)
Using \( m = 2 \) in the first equation:
\[ n = 11 - m = 11 - 2 = 9 \]

### Step 7: Solve for \( \sin x \) and \( \sec y \)
Now we substitute back to find \( \sin x \) and \( \sec y \):
1. \( m = 4^{\sin x} = 2 \) implies:
\[ 4^{\sin x} = 2 \]
\[ \sin x = \frac{1}{2} \]

2. \( n = 3^{\sec y} = 9 \) implies:
\[ 3^{\sec y} = 9 \]
\[ \sec y = 2 \]

### Step 8: Find Values of \( x \) and \( y \)
From \( \sin x = \frac{1}{2} \):
- \( x = n\pi + (-1)^n \frac{\pi}{6} \) for \( n \in \mathbb{Z} \)

From \( \sec y = 2 \):
- \( \cos y = \frac{1}{2} \) implies:
- \( y = 2m\pi + (-1)^m \frac{\pi}{3} \) for \( m \in \mathbb{Z} \)

### Final Answer
Thus, the exhaustive values of \( x \) and \( y \) respectively are:
- \( x = n\pi + (-1)^n \frac{\pi}{6} \)
- \( y = 2m\pi + (-1)^m \frac{\pi}{3} \)