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Differentiation Using Substitution

If α+β=π 4, t...

Question

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

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Solution

Given $α + β = \frac{π}{4}$
$(1 + \tan α) (1 + \tan β) 1 + \tan α + \tan β + \tan α \tan β = 1 + \tan (α + β) (1 - \tan α \tan β) + \tan α \tan β using identity : \tan (a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} = 1 + \tan (\frac{π}{4}) (1 - \tan α \tan β) + \tan α \tan β (∵ α + β = \frac{π}{4} given) = 1 + 1 (1 - \tan α \tan β) + \tan α \tan β = 1 + 1 - \tan α \tan β + \tan α \tan β = 2 Hence, (1 + \tan α) (1 + \tan β) = 2 Hence, the corrrect answer is option B . Since α + β = \frac{π}{4} \Rightarrow \tan (α + β) = \tan π / 4 \Rightarrow \frac{\tan α + \tan β}{1 - \tan α \tan β} = 1 (using identity : \tan (a + b) = \frac{\tan a + \tan b}{1 - \tan \tan b}) \Rightarrow \tan α + \tan β = 1 - \tan α \tan β \Rightarrow \tan α + \tan α \tan β + \tan β = 1 \Rightarrow 1 + \tan α + \tan β (1 + \tan α) = 1 + 1 \Rightarrow (1 + \tan α) (1 + \tan β) = 2 Hence, the correct answer option B .$

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