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प्रश्न
Find the general solution of the following equation:
उत्तर
Ideas required to solve the problem: The general solution of any trigonometric equation is given as:
sin x = sin y, implies x = nπ + (– 1)ny, where n ∈ Z.
cos x = cos y, implies x = 2nπ ± y, where n ∈ Z.
tan x = tan y, implies x = nπ + y, where n ∈ Z.
Given,
tan x + cot 2x = 0
⇒ tan x − cot 2x
We know that: cot θ = tan (π/2 − θ)
∴ `tan x = -tan (pi/2 - 2x)`
⇒ `tan x = tan (2x - pi/2) {∵ - tan θ = tan -θ}`
If tan x = tan y, then x is given by x = nπ + y, where n ∈ Z.
From above expression, on comparison with standard equation we have
y = `(2x - pi/2)`
∴ x = nπ + 2x − `pi/2`
⇒ `x = pi/2 - npi = pi/2(1 - 2n), "where" n ∈ Z`
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