Question

If A-B=π/4, then prove that (1+tanA)(1-tanB)=2​

Answer 1

Answer:

if A-B=π/4 then (1+TanA)(1-TanB)=2

Answer 2

Answer:

Proved that $(1+tanA)(1-tanB)=2$,

Step-by-step explanation:

Given to prove $(1+tanA)(1-tanB)=2$

Given $A-B=\frac{\pi }{4}$

$\implies A-B=\frac{\ 180 }{4}$

$\implies A-B=45^{o}$

Taking tan on both sides,

$tan(A-B)=1$

Using the trigonometric identity, $tan(A-B)=\frac{tanA-tanB}{1+tanAtanB}$, we get

$\frac{tanA-tanB}{1+tanAtanB} =1$

$\implies tanA-tanB=1+tanAtanB$                                             ...(1)

Consider the left hand side of the identity to be proved,

$(1+tanA)(1-tanB)$

Opening the brackets by multiplying the terms,

$=1+tanA-tanB-tanAtanB$

Substituting equation (1)

$=1+1+tanAtanB-tanAtanB$

$=2$

Hence, it is proved that $(1+tanA)(1-tanB)=2$,