Recall the formulas for changing coordinates between polar and rectangular coordinates: $x=rcos(),y=rsin(),$ and $r=\sqrt[\phantom{2}]{x^2+y^2}.$

$($a$)(1\mathrm{.}5$ points$)$ Using $r=\sqrt[\phantom{2}]{x^2+y^2},$ show that del$\frac{r}{d}$elx$=cos().$

$($b$)(1\mathrm{.}5$ points$)$ First, rewrite $x=rcos()$ as $r=\frac{x}{cos()}.$ Using this result, show that del$\frac{r}{d}$elx$=sec().$

$($c$)(5$ points$)$ Compare the results in $($a$)$ and $($b$).$ What's going on$...?!$ Explain the discrepancy$/$inconsistency CLEARLY and fix the problem.