Solve the following equations :
sin x tan x – 1 = tan x – sin x
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,
sin x tan x – 1 = tan x – sin x
⇒ sin x tan x – tan x + sin x – 1 = 0
⇒ tan x(sin x – 1) + (sin x – 1) = 0
⇒ (sin x – 1)(tan x + 1) = 0
∴ sin x = 1 or tan x = -1
⇒ sin x = sin π/2 or tan x = tan (- π/4 )
If sin x = sin y, implies x = nπ + (– 1)ny, where n ∈ Z.
and tan x = tan y, implies x = nπ + y, where n ∈ Z.
∴ x = nπ + (-1)n (π /2) or x = mπ + (- π/4)
∴