f(x) denotes the sum of infinite terms with of series
1- cosx +cos^2x - cos^3x + cos^4x —-----------------(x not equal to nπ)
& g(x) denotes the sum of infinite terms of series
1 + since + sin^2x + sin^3x + ————————(x is not equal to (2n+1)π/2)
Find all possible values are f x for which f(x) = g(x )
Answers
x = (4n - 1)π/4
Step-by-step explanation:
f(x) = 1 - Cosx + Cos²x - Cos³x + Cos⁴x - +.......................
this is an GP
a = 1
r = -Cosx
Sum = a/(1 - r) = 1/(1 - (-Cosx)) = 1/(1 + Cosx)
g(x) = 1 + Sinx + Sin²x + Sin³x + Sin⁴x - +.......................
this is an GP
a = 1
r = Sinx
Sum = a/(1 - r) = 1/(1 - Sinx)
f(x) = g(x)
=> 1/(1 + Cosx) = 1/(1 - Sinx)
=> 1- Sinx = 1 + Cosx
=> Cosx + Sinx = 0
=>Squaring both sides
=> Cos²x + Sin²x + 2SinxCosx = 0
=> 1 + Sin2x = 0
=> Sin2x = - 1
=> 2x = 2nπ - π/2
=> 2x = (4nπ - π)/2
=> x = (4n - 1)π/4
for x = (4n - 1)π/4 f(x) = gx
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