Reflection coefficient at the load when reflection coefficient at the mid point of transmission line given
The General formula of Reflection coefficient at the line is 

$$\Gamma_l(l)=\mid\Gamma_L\mid e^{j(\theta_\Gamma-2\beta l)}$$

at the mid point that is at \$l/2\$ the reflection coefficient is
$$\Gamma_l(l/2)=\mid\Gamma_L\mid e^{j(\theta_\Gamma-\beta l)}$$

Now \$\Gamma_l(l/2)\$ same as the \$\Gamma'\$ in the solution manual that is 
$$\Gamma_l(l/2)=0.4685\angle-38.666^{\circ}$$

Now I need to calculate the reflection coefficient at the load that is \$\mid\Gamma_L\mid e^{j(\theta_\Gamma)}\$=\$\Gamma_L\$ 

$$\Gamma_L=\Gamma_l(l/2)e^{\beta l}$$

Now calculate \$\beta l=\frac{2\pi\times300\times10^6\times4.2}{0.8\times3\times 10^8}=\frac{21 \pi}{2}=1890^{\circ}\$
$$\Gamma_L=\Gamma_l(l/2)e^{\beta l}=0.4685\angle-38.666e\times e^{j1890^{\circ}}$$
$$=0.4685\angle-38.666\times [cos(1890^{\circ})+jsin(1890)^{\circ}]$$
$$=0.4685\angle-38.666\times(0+j)$$ 
     where j is nothing but \$90^{\circ}\$

finally the answer becomes 

$$\boxed{0.4685\angle-38.666^{\circ}+90^{\circ}=0.4685\angle51.54^{\circ}}$$