If A+B=pi/4 show that [1+tan A][1+tanB]=2 or [1+cotA][1+cotB]=2cotAcotB

If A+B=pi/4 show that [1+tan A][1+tanB]=2  or  [1+cotA][1+cotB]=2cotAcotB

Given A+B=pi/4  or 45 degrees

 

tale tan on both sides

tan(A+B)=tan(pi/4)

using trigonometric formulae and standard values

{tanA+tanB} / {1-tanAtanB} =1

or

tanA +tanB =1 -tanAtanB

tanA +tanB +tanAtanB = 1

add 1 on both sides

1+tanA +tanB +tanAtanB = 1+1

1+tanA+tanB +tanAtanB = 2

factorise

1(1+tanA)+tanB(1+tanA)=2

 

  [1+tan A][1+tanB]=2

change tanA =1 / cotA  and tanB=1 /cotB in the above result and simplify to get the other expression

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