Question

if a+ b = 45° prove that (1+ tan a) ( 1+ tan b) =2

Answer 1

$\textbf{Given:}$

$A+B=45^{\circ}$

$\textbf{To prove:}$

$(1+\tan{A})(1+\tan{B})=2$

$\textbf{Solution:}$

$\text{Consider,}$

$A+B=45^{\circ}$

$\tan(A+B)=\tan{45^{\circ}}$

$\dfrac{\tan{A}+\tan{B}}{1-\tan{A}\tan{B}}=1$

$\tan{A}+\tan{B}=1-\tan{A}\tan{B}$

$\tan{A}+\tan{B}+\tan{A}\tan{B}=1$

$\text{Add 1 on bothsides, we get}$

$\tan{A}+\tan{B}+\tan{A}\tan{B}+1=2$

$\text{Factorize the L.H.S}$

$\tan{A}+\tan{A}\tan{B}+1+\tan{B}=2$

$\tan{A}(1+\tan{B})+1(1+\tan{B})=2$

$(\tan{A}+1)(1+\tan{B})=2$

$\implies\boxed{\bf(1+\tan{A})(1+\tan{B})=2}$

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