Let S = θ ∈ 0 , π 2 : ∑ m = 1 9 sec θ + m - 1 π 6 sec θ + m π 6 = - 8 3 . Then

Step-by-step solution

Let θ + m - 1 π 6 = x and θ + m π 6 = y So, y - x = π 6 Now, ∑ m = 1 9 sec θ + m - 1 π 6 sec θ + m π 6 = ∑ m = 1 9 sec x sec y = ∑ m = 1 9 1 cos x cos y = 2 ∑ m = 1 9 sin y - x cos x cos y = 2 ∑ m = 1 9 tan y - tan x = 2 ∑ m = 1 9 tan θ + m π 6 - tan θ + ( m - 1 ) π 6 = 2 tan θ + π 6 - tan θ + 0 · π 6 + 2 tan θ + 2 π 6 - tan θ + π 6 + 2 tan θ + 3 π 6 - tan &#952