Problem:
If A=20∘ and B=25∘, then the value of (1+tanA)(1+tanB) is
Answer Choices:
A. 3
B. 2
C. 1+2
D. 2(tanA+tanB)
E. none of these
Solution:
We show that for any angles A and B for which the tangent function is defined and A+B=45∘,(1+tanA)(1+tanB)=2. By the addition law for tangents,
1=tan45∘=tan(A+B)=1−tanAtanBtanA+tanB,1−tanAtanB=tanA+tanB,1=tanA+tanB+tanAtanB
Thus
(1+tanA)(1+tanB)=1+tanA+tanB+tanAtanB=1+1=2.