Question

If cos 2α = `(3 cos 2β - 1)/(3 - cos 2β)`, then tan α is equal to ______.

Options

• `sqrt(2) tan β`

• tan β

• sin 2 β

• `sqrt(2) cot β`

Answer 1

If cos 2α = `(3 cos 2β - 1)/(3 - cos 2β)`, then tan α is equal to `underlinebb(sqrt(2) tan β)`.

Explanation:

Given, cos 2α = `(3 cos 2β - 1)/(3 - cos 2β)`

`\implies (1 - tan^2 α)/(1 + tan^2 α) = (3((1 - tan^2 β)/(1 + tan^2 β)) - 1)/(3 - ((1 - tan^2 β)/(1 + tan^2 β))`

`\implies (1 - tan^2 α)/(1 + tan^2 α) = (2 - 4 tan^2 β)/(2 + 4 tan^2 β) = (1 - 2 tan^2 β)/(1 + 2 tan^2 β)`

Applying componendo and dividendo, we get

`1/(tan^2 α) = 1/(2 tan^2 β)`

`\implies` tan α = `sqrt(2) tan β`