Find the general solutions of the following equations :

sin 9x = 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,

⇒

Using transformation formula:

∴

⇒

∴ cos 5x = 0 or sin 4x = 0

If either of the equation is satisfied, the result will be 0

So we will find the solution individually and then finally combined the solution.

∴ cos 5x = 0

⇒ cos 5x = cos π/2

∴

,where n ϵ Z ………eqn 1

Also,

sin 4x = sin 0

Or ,where n ϵ Z ………eqn 2

From equation 1 and eqn 2,

or ,where n ϵ Z ...ans