4. (a) Solve \( \sec ^{2} x \tan y d x+\sec ^{2} y \tan x d y=0 \)

To solve the differential equation: \[ \sec^{2} x \tan y \, dx + \sec^{2} y \tan x \, dy = 0 \] we can proceed as follows: 1. **Check for Exactness:** Let \( M = \sec^{2} x \tan y \) and \( N = \sec^{2} y \tan x \). Compute the partial derivatives: \[ \frac{\partial M}{\partial y} = \sec^{2} x \cdot \sec^{2} y \] \[ \frac{\partial N}{\partial x} = \sec^{2} y \cdot \sec^{2} x \] Since \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\), the equation is **exact**. 2. **Find the Potential Function \( F(x, y) \):** Integrate \( M \) with respect to \( x \): \[ F(x, y) = \int \sec^{2} x \tan y \, dx = \tan y \int \sec^{2} x \, dx = \tan y \cdot \tan x + h(y) \] Now, differentiate \( F(x, y) \) with respect to \( y \) and equate it to \( N \): \[ \frac{\partial F}{\partial y} = \sec^{2} y \tan x + h'(y) = \sec^{2} y \tan x \] This implies \( h'(y) = 0 \), so \( h(y) \) is a constant. 3. **Write the General Solution:** \[ \tan x \tan y = C \] where \( C \) is an arbitrary constant. **Final Answer:** All solutions are given by tan x multiplied by tan y equals a constant. In symbols, tan x tan y = C