An optical fibre consists of core of ${\mathrm{\mu }}_1$surrounded by a cladding of ${\mathrm{\mu }}_2<{\mathrm{\mu }}_1.$A beam of light enters from air at an angle $\mathrm{\alpha }$ with axis of fibre. The highest $\mathrm{\alpha }$ for which ray can be travelled through fibre is

Minimum value of angle of  incidence at the core clad interface is critical angle between core and clad 

$\mathrm{sinC}=\frac{{\mathrm{\mu }}_2}{{\mathrm{\mu }}_1}$

So angle of refraction at the air core interface is r=90-C

$\mathrm{Then} \mathrm{sin\alpha }={\mathrm{\mu }}_1\mathrm{sinr}={\mathrm{\mu }}_1\sin \left(90-\mathrm{C}\right)={\mu }_1\mathrm{cosC}$

$\cos C=\sqrt{1-{\sin }^{2}C}=\sqrt{1-\frac{{\mu }_2^2}{{\mu }_1^2}}$

$\mathrm{sin\alpha }=$${\mu }_1\sqrt{1-\frac{{\mu }_2^2}{{\mu }_1^2}}=\sqrt{{\mu }_1^2-{\mu }_2^2}$