\(I=4 I_{0} \cos ^{2} \frac{\phi}{2}\)

where \(I_{0}\) is the intensity of either wave and \(\phi\) is the phase difference between two waves.

Phase difference, \(\phi=\frac{2 \pi}{\lambda} \times\) Path difference When path difference is \(\lambda,\) then

\(\phi=\frac{2 \pi}{\lambda} \times \lambda=2 \pi\)

\(\therefore \,\,I = 4{I_0}{\cos ^2}\left( {\frac{{2\pi }}{2}} \right)\) \( = 4{I_0}{\cos ^2}(\pi )\) \( = 4{I_0} = K\)    ....... \((i)\)

When path difference is \(\frac{\lambda}{4},\) then

\(\phi  = \frac{{2\pi }}{\lambda } \times \frac{\lambda }{4} = \frac{\pi }{2}\) 

\(\therefore \,\,{\text{ }}I = 4{I_0}{\cos ^2}\left( {\frac{\pi }{4}} \right)\)    \( = 2{I_0} = \frac{K}{2}\)   [using \((i)\)]