MathDB

In the adjoining figure of a rectangular solid,

\angle DHG=45^\circ

and

\angle FHB=60^\circ

. Find the cosine of

\angle BHD

.
[asy] size(200); import three;defaultpen(linewidth(0.7)+fontsize(10)); currentprojection=orthographic(1/3+1/10,1-1/10,1/3); real r=sqrt(3); triple A=(0,0,r), B=(0,r,r), C=(1,r,r), D=(1,0,r), E=O, F=(0,r,0), G=(1,0,0), H=(1,r,0); draw(D--G--H--D--A--B--C--D--B--F--H--B^^C--H); draw(A--E^^G--E^^F--E, linetype("4 4")); label("

A

", A, N); label("

B

", B, dir(0)); label("

C

", C, N); label("

D

", D, W); label("

E

", E, NW); label("

F

", F, S); label("

G

", G, W); label("

H

", H, S); triple H45=(1,r-0.15,0.1), H60=(1-0.05, r, 0.07); label("

45^\circ

", H45, dir(125), fontsize(8)); label("

60^\circ

", H60, dir(25), fontsize(8));[/asy]

\textbf {(A) } \frac{\sqrt{3}}{6} \qquad \textbf {(B) } \frac{\sqrt{2}}{6} \qquad \textbf {(C) } \frac{\sqrt{6}}{3} \qquad \textbf {(D) } \frac{\sqrt{6}}{4} \qquad \textbf {(E) } \frac{\sqrt{6}-\sqrt{2}}{4}

Problem Statement