Question

A+b=45 prove that (1+tana)(1+tanb)=2.Then what is the value of tan(45/2)

Answer 1

Answer:

(A+B)=45

multiply both sides with tan

so, tan(A+B)=tan45

tanA+tanB/1-tanAtanB=1

tanA + tanB = 1- tanAtanB

tanA +tanB + tanAtanB = 1...(1)

Now,given

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

1+tanB+tanA+ tanAtanB=2

tanA+anB+tanAanB=2-1

TanA+tanB+tanAtanB=1...(2)

Equation 1and 2 are equal

tan(45/2)=1

Answer 2

Given, A + B = 45°

⇒ tan(A + B) = tan45°

⇒ (tanA + tanB)/(1 - tanAtanB) = 1

⇒ tanA + tanB = 1 - tanAtanB

⇒ tanA + tanB + tanAtanB = 1

∴ (1 + tanA)(1 + tanB)

⇒ 1 + tanB + tanA + tanAtanB

⇒ 1 + 1

⇒ 2            proved

As we know, tan45° = 1, let tan(45/2)° be x.

=> 2x/(1 - x²) = 1

=> x² + 2x - 1 = 0

Using quadratic formula :

=> x = (- 2 ± √(2² - 4(-1)(1)) )/2(1)

=> x = (- 2 ± √8)/2 = (- 2 ± 2√2)/2

=> x = - 1 + √2 , as - 1 - √2 is -ve.

tan(45/2)° = √2 - 1