14. As shown in the figure, quadrilaterals $ABCD$ and $ADPQ$ are both squares, and the planes they lie in are mutually perpendicular. A moving point $M$ is on segment $PQ$. $\mathrm { E }$ and $\mathrm { F }$ are the midpoints of $\mathrm { AB }$ and $\mathrm { BC }$ respectively. Let the angle between skew lines $EM$ and $AF$ be $\theta$, then the maximum value of $\cos \theta$ is $\_\_\_\_$. [Figure]

13. As shown in the figure, in the triangular pyramid $A - B C D$, $AB = AC = BD = CD = 3$ , $AD = BC = 2$ , and $M , N$ are the midpoints of $AD , BC$ respectively. Then the cosine of the angle between the skew lines $AN$ and $CM$ is $\_\_\_\_$ .

10. In a right triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$, $\angle A B C = 120 ^ { \circ } , A B = 2 , B C = C C _ { 1 } = 1$. The cosine of the angle between skew lines $A B _ { 1 }$ and $B C _ { 1 }$ is
A. $\frac { \sqrt { 3 } } { 2 }$
B. $\frac { \sqrt { 15 } } { 5 }$
C. $\frac { \sqrt { 10 } } { 5 }$
D. $\frac { \sqrt { 3 } } { 3 }$ [Figure]

In the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, $E$ is the midpoint of edge $C C _ { 1 }$. The tangent of the angle between skew lines $A E$ and $C D$ is
A. $\frac { \sqrt { 2 } } { 2 }$
B. $\frac { \sqrt { 3 } } { 2 }$
C. $\frac { \sqrt { 5 } } { 2 }$
D. $\frac { \sqrt { 7 } } { 2 }$

In rectangular prism $A B C D - A _ { 1 } B C _ { 1 } D _ { 1 }$, $A B = B C = 1 , A A _ { 1 } = \sqrt { 3 }$, the cosine of the angle between skew lines $A D _ { 1 }$ and $D B _ { 1 }$ is
A. $\frac { 1 } { 5 }$
B. $\frac { \sqrt { 5 } } { 6 }$
C. $\frac { \sqrt { 5 } } { 5 }$
D. $\frac { \sqrt { 2 } } { 2 }$

At the transition between the two sections of the borehole, the drilling direction must be changed by the angle that is described in the model by the angle of intersection of the two lines $A P$ and $P Q$. Determine the size of this angle.