Neerav,

This works for me -- you have to remove the Reals because Solve can't know about your parameters without assumptions.

eq1 = ExpandAll[Mm^2 + Ll^2 - 2 Mm Ll Cos[\[Phi] i] == a]
eq2 = ExpandAll[Mm^2 + Xx^2 - 2 Mm Xx Cos[\[Phi] c] == b]
eq3 = ExpandAll[Ll^2 + Xx^2 - 2 Ll Xx Cos[\[Phi] i - \[Phi] c] == c]

Solve[{eq1, eq2, eq3}, {Ll, Mm, Xx}]

For any two of your three equations and two of your three variables

eq1 = M^2 + L^2 - 2 M L Cos[i] == a
eq2 = M^2 + X^2 - 2 M X Cos[cr] == b
eq3 = L^2 + X^2 - 2 L X Cos[i - cr] == c
Reduce[{eq2, eq3}, {L, M}]

it almost instantly returns a solution in terms of the remaining variables.

Is there any chance that you might more quickly be able to take the result of two of the two variable solutions and equate those find the overall solution and then verify your condition that the result be Real?

Is i supposed to be the unit imaginary number? If so, it needs to be I. See this.

Hello Nareev,

"Phii" and "Phic" need spaces or "*": "Phi i" and "Phi c" OR "Phi*i" and "Phi*C" Also, "cos" needs to be "Cos" because Cosine is a built-in function of the Wolfram Language.