In the adjacent diagram,CP represents a wavefront and AO and BP, the corresponding two rays. Find the condition on $\mathrm{\theta }$ for constructive interference at P between the ray BP and reflected rays OP

PR=d $\mathrm{PO}=\mathrm{dsec}\mathrm{\theta },$, and $\mathrm{CO}=\mathrm{POcot}2\mathrm{\theta }=\mathrm{dsec}\mathrm{\theta cos}2\mathrm{\theta }$ Optical path difference between the two ras is $\mathrm{\Delta x}=\mathrm{CO}+\mathrm{PO}=\mathrm{dsec}\mathrm{\theta }+\mathrm{dsec}\mathrm{\theta cos}2\mathrm{\theta }+\frac{\mathrm{\lambda }}{2}$
Path difference between the two rays is
$\mathrm{\Delta \varphi }=2\mathrm{n\pi },\mathrm{n}=0,1,2,3\dots$
Condition for constructive interference should be
……
$\begin{array}{l}\mathrm{\Delta x}=\mathrm{n\lambda },\mathrm{n}=0,1,2,3\dots .\\ \mathrm{dsec}\mathrm{\theta }[1+\cos 2\mathrm{\theta }]=\frac{\mathrm{\lambda }}{2}\left(\frac{\mathrm{d}}{\cos \mathrm{\theta }}\right)\left(2{\cos }^2\mathrm{\theta }\right)=\frac{\mathrm{\lambda }}{2}\\ \cos \mathrm{\theta }=\frac{\mathrm{\lambda }}{4\mathrm{d}}\end{array}$