MathDB

A circle centered at

O

has radius

1

and contains the point

A

. Segment

AB

is tangent to the circle at

A

and \angle{AOB} \equal{} \theta. If point

C

lies on

\overline{OA}

and

\overline{BC}

bisects

\angle{ABO}

, then OC \equal{}
[asy]import olympiad; unitsize(2cm); defaultpen(fontsize(8pt)+linewidth(.8pt)); labelmargin=0.2; dotfactor=3;
pair O=(0,0); pair A=(1,0); pair B=(1,1.5); pair D=bisectorpoint(A,B,O); pair C=extension(B,D,O,A);
draw(Circle(O,1)); draw(O--A--B--cycle); draw(B--C);
label("

O

",O,SW); dot(O); label("

\theta

",(0.1,0.05),ENE);
dot(C); label("

C

",C,S);
dot(A); label("

A

",A,E);
dot(B); label("

B

",B,E);[/asy]
(A)\ \sec^2\theta \minus{} \tan\theta \qquad (B)\ \frac {1}{2} \qquad (C)\ \frac {\cos^2\theta}{1 \plus{} \sin\theta} \qquad (D)\ \frac {1}{1 \plus{} \sin\theta} \qquad (E)\ \frac {\sin\theta}{\cos^2\theta}

17

Problems(1)