Solve the equations for value of x

tan mx + cot nx = 0

This will be true if

nx = -[pi/2 - mx]
= mx - pi/2
(m - n)x = pi/2
x = (pi/2)/(m-n)

Check: suppose m = 3 and n = 1
Then x = pi/4

tan 3pi/4 =? -cot pi/4
-1 = -1

To solve the equation tan(mx) + cot(nx) = 0 for the value of x, we can follow these steps:

Step 1: Rewrite the equation using trigonometric identities.
tan(mx) + cot(nx) = 0
sin(mx)/cos(mx) + cos(nx)/sin(nx) = 0

Step 2: Find a common denominator for the fractions.
(sin(mx)sin(nx) + cos(mx)cos(nx))/(cos(mx)sin(nx)) = 0

Step 3: Simplify the numerator.
cos(mx - nx) = 0

Step 4: Solve for x.
mx - nx = (2k + 1) * π/2 (where k is an integer)
(m - n)x = (2k + 1) * π/2
x = (2k + 1) * π/2 / (m - n)

So, the solution for x is x = (2k + 1) * π/2 / (m - n), where k is an integer.