Find the number solution are ordered pair `(x,y)` of the equation` 2^(sec^(2)x)+2^("cosec"^(2)y)=2cos^(2)x(1-cos^(2)y)` in ` [0,2pi]`

Answer

To solve the equation \( 2^{\sec^2 x} + 2^{\csc^2 y} = 2 \cos^2 x (1 - \cos^2 y) \) in the interval \([0, 2\pi]\), we will analyze both sides of the equation step by step.

### Step 1: Analyze the Right-Hand Side (RHS)
The RHS of the equation is given by:
\[ \text{RHS} = 2 \cos^2 x (1 - \cos^2 y) \]
We know that \( \cos^2 x \) and \( \cos^2 y \) both lie between 0 and 1. Therefore, \( 1 - \cos^2 y \) also lies between 0 and 1. Thus, the maximum value of the RHS occurs when both \( \cos^2 x \) and \( 1 - \cos^2 y \) are at their maximum values.

### Step 2: Find Maximum Value of RHS
The maximum value of \( \cos^2 x \) is 1 (when \( x = 0, \pi \)), and the maximum value of \( 1 - \cos^2 y \) is also 1 (when \( y = \frac{\pi}{2} \)). Therefore, the maximum value of the RHS is:
\[ \text{RHS}_{\text{max}} = 2 \cdot 1 \cdot 1 = 2 \]

### Step 3: Analyze the Left-Hand Side (LHS)
The LHS of the equation is:
\[ \text{LHS} = 2^{\sec^2 x} + 2^{\csc^2 y} \]
The functions \( \sec^2 x \) and \( \csc^2 y \) are always greater than or equal to 1 for all \( x \) and \( y \) in the given interval. Therefore, the minimum value of \( \text{LHS} \) occurs when both \( \sec^2 x \) and \( \csc^2 y \) are equal to 1.

### Step 4: Find Minimum Value of LHS
The minimum value of \( \sec^2 x \) is 1 (when \( x = 0, \pi \)), and the minimum value of \( \csc^2 y \) is also 1 (when \( y = \frac{\pi}{2} \)). Therefore, the minimum value of the LHS is:
\[ \text{LHS}_{\text{min}} = 2^1 + 2^1 = 2 + 2 = 4 \]

### Step 5: Compare LHS and RHS
From our analysis:
- The maximum value of the RHS is 2.
- The minimum value of the LHS is 4.

Since the minimum value of the LHS (4) is greater than the maximum value of the RHS (2), there are no values of \( x \) and \( y \) that satisfy the equation.

### Conclusion
Thus, the number of ordered pairs \( (x, y) \) that satisfy the equation in the interval \([0, 2\pi]\) is:
\[ \boxed{0} \]