Verified answer

5∙cot(2∙(x - 60)) + √2 = √2

5∙cot(2x - 120) = 0

cot(2x - 120) = 0

cos(2x - 120)

-------------------- = 0

sin(2x - 120)

cos(2x - 120) = 0

2x - 120 = 90 + k∙180

2x = 210 + k∙180

x = 105 + k∙90

Edit:

corrected after your Additional Details

5 * cot(x - 60)^2 + sqrt(2) = sqrt(2)

5 * cot(x - 60)^2 = 0

cot(x - 60)^2 = 0

cot(x - 60) = 0

cos(x - 60) / sin(x - 60) = 0

cos(x - 60) = 0

x - 60 = 90 + 180 * k

x = 150 + 180 * k

k is an integer