If α + β = (π)/2 and β + γ = α, then tanα equals hspace2em — JEE Main 2001 Mathematics PYQ

2001

\alpha + \beta = \dfrac{\pi}{2}

and

\beta + \gamma = \alpha

, then

\tan\alpha

equals

\hspace{2em}

A.
$2(\tan\beta + \tan\gamma)$
B.
$\tan\beta + \tan\gamma$
C.
$\tan\beta + 2\tan\gamma$
D.
$2\tan\beta + \tan\gamma$

Show answer and solution

Correct option: C

Solution

From α + β = π/2, we get α = π/2 - β, so tan α = cot β.

From β + γ = α, we get γ = α - β, so tan γ = tan(α - β) = (tan α - tan β)/(1 + tan α tan β).

Since tan α = cot β = 1/tan β, we have tan α · tan β = 1, making the denominator 1 + 1 = 2.

So tan γ = (tan α - tan β)/2, which gives 2tan γ = tan α - tan β, hence tan α = tan β + 2tan γ.

[2001]

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