Question

If sin (x + 3 α) = 3 sin (α – x), then ______.

Options

• tan x = tan α

• tan x = tan² α

• tan x = tan³ α

• tan x = 3 tan α

Answer 1

If sin (x + 3 α) = 3 sin (α – x), then tan x = tan³ α.

Explanation:

Given, `(sin(x + 3α))/(sin(α - x))` = 3

On applying componendo and dividendo rule, we get

`(sin(x + 3α) + sin(α - x))/(sin(x + 3α) - sin(α - x)) = (3 + 1)/(3 - 1)`

`\implies (2 sin 2 α cos (α + x))/(2 cos 2 α sin (α + x))` = 2

`\implies (tan 2 α)/(tan(α + x))` = 2

`\implies (2 tan α)/(1 - tan α) xx ((1 - tan α tan x))/((tan α + tan x))` = 2

`\implies` tan α – tan² α tan x = tan α + tan x – tan³ α – tan² α tan x

`\implies` tan x = tan³ α