In a previous post I asked (Which theorems have Pythagoras' Theorem as a special case?).
Are there any compelling examples where it is significantly "easier"/"simpler" to prove a more general or more advanced theorem with the purpose of deriving the desired theorem as a special case?
For example, imagine you have a super-simple proof of Brahmagupta's formula: the area $K$ of a cyclic quadrilateral whose sides have lengths $a$, $b$, $c$, $d$ is $$K={\sqrt {(s-a)(s-b)(s-c)(s-d)}},$$ where $s$, the semiperimeter, is defined to be $s=\frac{a+b+c+d}{2}$. Then set $d=0$ to derive Heron's formula.
Yes, simple direct proofs of Heron exist (and I have a small collection), but the point is to illustrate the idea of proof by generalising. In this example, to prove Heron we generalise to Brahmagupta.
Are there any really compelling examples where the known direct proofs of the desired theorem are significantly more complex than the simple proof of the more general result?
