If a=12 and (a+1)(b+1)=2, then the radian measure of Arctana+Arctanb equals

1979 AMC 12 Problem 20

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20.

a=\frac12

and

(a+1)(b+1)=2,

then the radian measure of

\operatorname{Arctan}a+\operatorname{Arctan}b

equals

\frac\pi2

\frac\pi3

\frac\pi4

\frac\pi5

\frac\pi6

First solve the given product equation for

b

Use the tangent addition formula on the two positive inverse-tangent angles

Substituting

a=\frac{1}{2}

gives

b=\frac{1}{3}.

u=\operatorname{Arctan}a

and

v=\operatorname{Arctan}b,

then

\begin{aligned} \tan(u+v)&=\frac{a+b}{1-ab}\\ &=\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{6}}\\ &=1. \end{aligned}

Both angles are positive and their sum is less than

\frac{\pi}{2},

u+v=\frac{\pi}{4}.

Therefore, the correct answer is C.