The path difference between the two wave $y_1=a_1\sin (\omega t-\frac{2\pi x}{\lambda })$  and $y_2=a_2\cos (\omega t-\frac{2\pi x}{\lambda }+\varphi )$ is

$$\begin{gathered} y_2=a_2\cos [\omega t-\frac{2\pi x}{\lambda }+\varphi ] y_2=a_2\sin [\frac{\pi }{2}+(\omega t-\frac{2\pi x}{\lambda }+\varphi )] \\ {y}_1=a_1\sin (\omega t-\frac{2\pi x}{\lambda }+\varphi ) \end{gathered}$$

Phase difference = $\frac{\pi }{2}+\varphi$

Path difference $=\frac{\lambda }{2\pi }\times$  phase difference $=\frac{\lambda }{2\pi }[\frac{\pi }{2}+\varphi ]$