Is the law of cosines the only equation that expresses the relations between the sides and angles of a triangle? (Other than the law of sines). For example: Suppose we have a triangle and sides a,b and the angle opposite of c is known. Then invoking the law of cosines, we know that c sq = a sq + b sq - 2abcos(angle). Is this the only valid expression that exists that tells us the value of c sq? Is it possible that other expressions, for example, (just making this up) that c sq = b * pi - cos(4 * pi) + a sq also gives us the same number determined by the law of cosines? Or, is the law of cosines the only valid expression giving us the correct answer. Has it been proven rigorously in mathematics or can it be proven that this the only unique expression that exists that gives us the right answer? And if this is the case, if an alternative expression is found that gives us the same value of c sq for some given values of a,b and angle, can this only occur if the alternative expression can be algebraically derived from the law of cosines?