Question

If α + β = `π/2` and β + γ = α, then the value of tan α is ______.

Options

• tan β + tan γ

• 2 (tan β + tan γ)

• tan β + 2 tan γ

• 2 tan β + tan γ

Answer 1

If α + β = `π/2` and β + γ = α, then the value of tan α is tan β + 2 tan γ.

Explanation:

Given, α + β = `π/2` and β + γ = α

`\implies` β = `π/2 - α` and β + γ = α

`\implies` tan β = `tan (π/2 - α)`

`\implies` tan β = cot α  ...(i)

and tan (β + γ) = tan α

`\implies (tan β + tan γ)/(1 - tan β tan γ)` = tan α

`\implies (tan β + tan γ)/(1 - cot α tan γ) = tanα/1`  ...[from equation (i)] 

`\implies` tan β + tan γ = tan α – tan γ

`\implies` tan α = tan β + 2 tan γ