Answer
$29.7^\circ$
Work Step by Step
First of all, we need to find the second-order angles ($m=2$) for the two given wavelengths.
We can use the formula of
$$d\sin\theta =m\lambda $$
Solving for $\theta$;
$$ \theta =\sin^{-1}\left[\dfrac{m\lambda }{d}\right]$$
Now we need to find $d$ which is given by $d=\dfrac{1}{6.5\times10^5}$
Thus,
$$ \theta =\sin^{-1}\left[ 6.5\times10^5m\lambda \right]$$
For $\lambda_1=7\times 10^{-7}$ m,
$$ \theta_1 =\sin^{-1}\left[ 6.5\times10^5\times 2\times 7\times 10^{-7} \right]=\bf 65.5^\circ$$
For $\lambda_2=4.5\times 10^{-7}$ m,
$$ \theta_2 =\sin^{-1}\left[ 6.5\times10^5\times 2\times 4.5\times 10^{-7} \right]=\bf 35.8^\circ$$
Therefore, the needed angle separation is given by
$$\Delta \theta=\left| \theta_2-\theta_1\right|=|35.8^\circ- 65.5^\circ|=\color{red}{\bf 29.7^\circ}$$
