Question:

Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. From $\dfrac{4}{2+\sqrt{4-\sqrt{c}}}=\sqrt{2}$, cross-multiply: $2+\sqrt{4-\sqrt{c}}=\dfrac{4}{\sqrt{2}}=2\sqrt{2}$. So $\sqrt{4-\sqrt{c}}=2\sqrt{2}-2$. Squaring: $4-\sqrt{c}=8-8\sqrt{2}+4$. Solving gives $c=2$. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.