Find the general solutions of the following equations :
tan mx + cot nx = 0
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,
⇒
We know that: cot θ = tan (π/2 – θ)
∴
⇒ { ∵ - tan θ = tan -θ }
If tan x = tan y, then x is given by x = kπ + y, where k ∈ Z.
From above expression, on comparison with standard equation we have
y =
∴
⇒
,where k ϵ Z …ans