If an orthogonal coordinate system transforms to another system of coordinates which is not orthogonal, then does the relation between the direction cosines in the orthogonal system, i.e. $l^2+m^2+n^2=1$ still resists in the other transformed system??
Please tell me if my question is wrong.
There is a doubt!!
"Orthogonal" means that the basis vectors for the coordinate system are all perpendicular to each other. I.e., that $l_1l_2 + m_1m_2 + n_1n_2 = 0$. It does not imply that $l_i^2+m_i^2+n_i^2=1$. That condition is called being "normal". When both hold, the coordinate sytem is "orthonormal".
There is no requirement that coordinate systems have to be orthogonal or normal. So no, When transforming an orthonormal coordinate system to another coordinate system, you cannot depend on the new system having either property.