Question

If tan X + cot X = 2, then find sin nth X + cos nth X.

Answer 1

tanx + 1/tanx = 2
tan^2 x - 2 tanx +1 = 0
=>  tan x = 1
=>  x = π/4 + m π   , where m = integer.
we take only the prime solution.  x = π/4

Let n = integer.
sin nx + cos n x
   = sin (n π/4) + cos (nπ/4)
   = Sin (n π/4) + Sin (π/2 - n π/4)
   = 2 Sin [ π/4] * Cos [ (n-1) π/4 ]
   = √2 Cos [(n-1) π/4]

Ans  = 1    n = 0
      = √2    n = 1
      = 1      n = 2
      = 0      n = 3
      = -1     n = 4
      = -√2   n = 5
      = -1     n = 6
      = 0      n = 7
      = 1      n = 8
and so on..