F (x) = sin (n π - x) cos (n π + x) / cos ((n + 1) π - x) * Tan (x-n π) * cot (n π / 2 + x), find the value of F (π / 6)

F (x) = sin (n π - x) cos (n π + x) / cos ((n + 1) π - x) * Tan (x-n π) * cot (n π / 2 + x), find the value of F (π / 6)

If n is even f (x) = sin (n π - x) cos (n π + x) / cos ((n + 1) π - x) * Tan (x-n π) * cot (n π / 2 + x) = sin (- x) cosx / [- cos (- x)] tanxcotx = SiNx = sin π / 6 = 1 / 2, if n is odd f (x) = sin (n π - x) cos (n π + x) / cos ((n + 1) π - x) * Tan (x-n π) * cot (n π / 2 + x) = [- sin

π given f (x) = sin (n π - x) cos (n π + x) / cos [(n + 1) π - x] * Tan (x-n π) (n belongs to Z) find f (6 / 7 π)

f(x)=sin(nπ-x)cos(nπ+ x)/cos[(n+1)π-x]*tan(x-nπ)
=sin(nπ-x)cos(nπ+ x)/[-cos(nπ-x)]*[-tan(nπ-x)]
=cos(nπ+ x)
f(7π/6)=cos(nπ+7π/6)
=±√3/2

It is known that f (α) = (Tan (π - α) * cos (2 π - α) * sin (half π + α)) / (COS (- α - π)) (1) simplify f (α)

F (α) = (Tan (π - α) * cos (2 π - α) * sin (half π + α)) / (COS (- α - π))
=-tana*cos(-α)*cosa/(-cosa)
=tana*cosα*cosa/cosa
=sina
Using induced formula, odd variable and even constant, looking at quadrant with sign

Given f (x) = [sin (- n π - x) cos (n π + x)] / cos [(n + 1) π - x] × 1 / Tan (x-n π) (n ∈ z), find f (7 π / 6)

① When n = 2K,
f(x) =-sinxcosx/(-cosx)·(1/tanx)
=-cosx
f(7π/6)=√3/2
② When n = 2K + 1,
f(x)=sinx·(-cosx)/cosx·(1/tanx)
=-cosx
f(7π/6)=√3/2
When n ∈ Z, f (7 π / 6) = √ 3 / 2