If x belongs to [0,2π] then number of values of X satisfying the equation tan8x = (cos x - sin x)/(cos x + sin x) is
Answers
Given : x ∈ [0,2π] , tan8x = (cos x - sin x)/(cos x + sin x)
To Find : number of values of x satisfying the equation
Solution:
Tan8x = (cosx - Sinx)/(Cosx + Sinx)
using Sinx = Cosx.Tanx
=>Tan8x = Cosx(1 - Tanx)/Cosx(1 + Tanx)
=> Tan8x = (1 -Tanx)/(1 + Tanx)
=> Tan8x = (1 -Tanx)/(1 + 1.Tanx)
1 = Tan(nπ+ π/4) n ∈ Z
=> Tan8x = (Tan(nπ+π/4) -Tanx)/(1 + Tan(nπ+π/4).Tanx)
Tan(A - B) = (Tan A - TanB)/(1 + TanA.TanB)
=> Tan8x = Tan(nπ+π/4 - x)
8x = nπ + π/4 - x
=> 9x = nπ + π/4
=> x = (4nπ + π)/36
x ∈ [0,2π]
0 ≤ (4nπ + π)/36 ≤ 2π
=> 0 ≤ (4nπ + π) ≤ 72π
=> 0 ≤ 4nπ + π ≤ 72π
=> 0 ≤ 4nπ + π ≤ 72π
=> -π ≤ 4nπ ≤ 71π
=> -1 ≤ 4n ≤ 71
=> -1/4 ≤ n ≤ 71/4
=> n = 0 to 17
18 possible values
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