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Suppose i have to calculate the ratio of lengths of AB and CD where A(0° N,0° E) and B(0°N,30° E), and C(30°N,0°E) and D(30°N,30° E) the values are just as an example.Is it possible to get the ratio in terms of theta(latitude of AB) and phi(latitude CD) and not having radius of earth in equation(which i suppose should be cancelled when we take ratio)

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Assuming the Earth is a sphere (which is a good approximation) radius $R$, and assuming you want great-circle distances (which you should) then given two points with latitude and longitude $\phi_1, \theta_1$, $\phi_2, \theta_2$ then the great-circle distance between them is

$$R \cos^{-1}\left(\sin\phi_1 \sin\phi_2 + \cos\phi_1 \cos\phi_2 \cos\left(\left|\theta_2 - \theta_1\right|\right)\right) $$

This formula is not numerically brilliant for small distances I think, although it's OK so long as your floats are long enough.

It's easy to see that $R$ cancels if you want only the ratio of distances.

You could have looked this up rather easily: Wikipedia for instance has this formula under 'great-circle distance'.

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It is like the Thales relation for triangle sides. If $P$ be the north pole then $$\dfrac{PC}{PA}=\dfrac{CD}{AB}$$ since the points locate on pure shpere then arc form is $$\dfrac{60^\circ}{90^\circ}=\dfrac{CD}{AB}=\dfrac23$$

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