Question

a+b=45,tanA+tanB+tanA×tanB=1

Answer 1

Given A+B=45  {Take tan on both the sides }
tan(A+B) = tan45 
tanA+tanB/1- tanA tanB = 1
tanA+tanB=1-tanA.tanB
tanA+tanB+tanA.tanB=1
adding "1" on both sides
1+ tanA+tanB+tanA.tanB=1+1
(1 + tanA)+tanB(1+tanA).=2
(1+tanA)(1+tanB)=2   Hence proved .
hope it's help u

Answer 2

Hey !!!

A + B = 45° -------(given)

tan(A +B) = tan45°

tanA + tanB/1-tanA×tanB = 1
•°• tan(A+B) = tanA + tanB/1-tanA×tanB

tanA + tanB = 1 - tanA×tanB

tanA + tanB + tanA×tanB = 1 {prooved}

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Hope it helps you !!!.

@Rajukumar111