Angular width of Refracted Beam!

I don't think the answer given is exactly right, but here is what I got:

From Snell's law:

$\frac { \sin { \theta } }{ \sin { { \theta }_{ 1 } } } =\quad n(\lambda )$

$\Rightarrow \frac { \sin { \theta } }{ n(\lambda ) } =\sin { { \theta }_{ 1 } } ........(1)$

Taking derivative on both sides:

$\Rightarrow \frac { -\sin { \theta } }{ { (n(\lambda )) }^{ 2 } } \times dn(\lambda )\times \delta \lambda =\cos { { \theta }_{ 1 } } \times \delta { \theta }_{ 1 }........(2)$

Divide $(2)$ by $(1)$ :

$\Rightarrow \frac { -dn(\lambda )\times \delta \lambda }{ n(\lambda ) } =\cot { { \theta }_{ 1 } } \times \delta { \theta }_{ 1 }$

Neglect minus sign:

$\Large \boxed{\delta { \theta }_{ 1 }=\frac { \tan { { \theta }_{ 1 } } \times dn(\lambda ) }{ n(\lambda ) } \delta \lambda }$