Solve the given differential equation by separation of variables. `sec^2 xdy + cscydx = 0`

Solve the given differential equation by separation of variables.    `sec^2 xdy + cscydx = 0`

Solution:

To solve this differential equation by separation of variables, we start by dividing both sides by `sec^2 x` and multiplying both sides by dx:

`dy/dx + (cscy/sec^2 x) dx = 0`

Next, we move all the terms involving dx to one side and all the terms involving dy to the other side:

`dy/dx = -(cscy/sec^2 x) dx`

Now we can integrate both sides with respect to their respective variables:

`∫ dy = -∫ (cscy/sec^2 x) dx`

`y = -cscy ln |sec x| + C`

where C is an arbitrary constant.

So the solution to the differential equation is:

`y = -cscy ln |sec x| + C`