Answer
Proofs given below.
Work Step by Step
${\bf v}=\langle a,b,c\rangle$
The angle between ${\bf i}$ and ${\bf v}$ is
$\displaystyle \cos\alpha=\cos\angle({\bf i,v})=\frac{{\bf i}\cdot{\bf v}}{|{\bf i}||{\bf v}|}$,
Similarly for $\beta$ and $\gamma$, replace ${\bf i}$ with ${\bf j}$ and ${\bf k}$.
${\bf (a)}$
$\displaystyle \cos\alpha=\frac{(1)(a)+(0)(b)+(0)(c)}{1\cdot|{\bf v}|}=\frac{a}{|{\bf v}|}$
$\displaystyle \cos\beta=\frac{{\bf j}\cdot{\bf v}}{|{\bf j}||{\bf v}|}=\frac{(0)(a)+(1)(b)+(0)(c)}{1\cdot|{\bf v}|}=\frac{b}{|{\bf v}|}$
$\displaystyle \cos\gamma=\frac{{\bf k}\cdot{\bf v}}{|{\bf k}||{\bf v}|}=\frac{(0)(a)+(0)(b)+(1)(c)}{1\cdot|{\bf v}|}=\frac{c}{|{\bf v}|}$
${\bf (b)}$
If $|{\bf v}|=1$, from the above results, it follows that
$\displaystyle \cos\alpha=\frac{a}{|{\bf v}|}=a,$
$\displaystyle \cos\beta=\frac{b}{|{\bf v}|}=b$
$\displaystyle \cos\gamma=\frac{c}{|{\bf v}|}=c$
( a, b, and c are the direction cosines of ${\bf v}$).
