If α + β = π/4, then the value of (1 + tan α)(1 + tan β) is (a) 1 (b) 2 (c) –2 (d) not defined

If α + β = π/4, Find the Value of (1 + tan α)(1 + tan β)

Question:
If \[ \alpha+\beta=\frac{\pi}{4} \] then the value of \[ (1+\tan\alpha)(1+\tan\beta) \] is

(a) \(1\)
(b) \(2\)
(c) \(-2\)
(d) not defined

Solution

Using the tangent addition formula:

\[ \tan(\alpha+\beta) = \frac{ \tan\alpha+\tan\beta } { 1-\tan\alpha\tan\beta } \]

Given,

\[ \alpha+\beta=\frac{\pi}{4} \]

Therefore,

\[ \tan\left(\frac{\pi}{4}\right)=1 \]

Hence,

\[ \frac{ \tan\alpha+\tan\beta } { 1-\tan\alpha\tan\beta } =1 \]

Cross multiplying,

\[ \tan\alpha+\tan\beta = 1-\tan\alpha\tan\beta \]

Adding \(1+\tan\alpha\tan\beta\) to both sides,

\[ 1+\tan\alpha+\tan\beta+\tan\alpha\tan\beta=2 \]

Factorizing,

\[ (1+\tan\alpha)(1+\tan\beta)=2 \]

\[ \boxed{ (1+\tan\alpha)(1+\tan\beta)=2 } \]

Correct Option: (b)

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