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  4. If A+B=(5π/4), then value of (1 + tan A)(1 + tan B) equals

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Q. If $A+B=\frac{5\pi}{4}$, then value of $\left(1 + tan\, A\right)\left(1 + tan \,B\right)$ equals

Trigonometric Functions

Solution:

$tan\left(A+B\right)=tan\, \frac{5\pi}{4}$
$=tan\left(\pi+\frac{\pi}{4}\right)=tan \frac{\pi}{4}$
$\therefore tan\, A + tan \,B = 1 - tan \,A\, tan \,B$
$\Rightarrow tan \,A + tan \,B + tan\, A \,tan\,B = 1$
$\Rightarrow \left(1 + tan\, A\right) + tan\,B \left(1 + tan\, A\right) = 1 + 1$
(by adding $1$ on both sides)
$\Rightarrow (1 + tan \,A) (1 +tan\,B) = 2$