Can anyone here please explain how to notice Aczel's identity in a certain question. I mean in questions involving inequalities, on understanding the inequality one gets a hint to apply a certain inequality e. g. Cauchy Schwarz, Jensen's inequality, Chebyshev's inequality, etc. So how does a question involving Aczel's inequality be identified.
Eg. In this question how does one get to know that this question uses Aczel's identity.
Suppose $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ are real numbers such that $$(a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 -1)(b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 - 1) > (a_1 b_1 + a_2 b_2 + \cdots + a_n b_n - 1)^2.$$Prove that $a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 > 1$ and $b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 > 1$
Note: If someone wishes he/she can post the solution to this problem.