Question

Find the general solution of the following equation:

\[\tan x + \cot 2x = 0\]

Answer 1

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`