Cot2x - tan 78 = ( sec x sec 78 )/2

Here's another solution

I showed this to Rod and he corrected an error in the final steps, The full solution is as follows.

I let y = 78 degrees in the expression above and simplified it from there.

cot(2x) – tan(y) = ( sec(x).sec(y) )/2

cos(2x)/sin(2x) – sin(y)/cos(y) = 1/(2cos(x).cos(y))

{cos(2x).cos(y) – sin(2x).sin(y)}/{sin(2x).cos(y)} = 1/(2cos(x).cos(y))

{cos(2x+y)}/{sin(2x).cos(y)} = 1/(2cos(x).cos(y))

{cos(2x+y)}/{sin(2x)} = 1/(2cos(x))

cos(2x+y) = sin(2x)/(2cos(x))

cos(2x+y) = 2sin(x).cos(x)/(2cos(x))

cos(2x+y) = sin(x)

cos(2x+y) = cos(pi/2-x)

equating the arguments of the two cosines,

2x + y = pi/2 – x + 2n.pi

3x = pi/2 + 2n.pi – y

3x = 90 + 360n – 78

3x = 12 + 360n

x = 4 + 120n

**x = 4 + 0, 120, 240, 360, …**