Solve the following equations :

3 tan x + cot x = 5 cosec 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,

3tan x + cot x = 5cosec x

⇒

⇒

⇒

⇒

⇒ {∵ sin² x + cos² x = 1}

∴

⇒ = 0

⇒

⇒

⇒

∴ cos x = -3 (neglected as cos x lies between -1 and 1)

or cos x = � (accepted value)

∴

If cos x = cos y, implies x = 2nπ ± y, where n ∈ Z.

∴