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An implicit equation specifies a relation between variables without isolating a designated output as an explicit function of the inputs. For example, $x^2+y^2=1$ specifies a circle. On $-1<x<1$, it determines the two real branches $y=\pm \sqrt{1-x^2}$, each an implicit function when one branch is selected. The implicit function theorem gives local conditions for a unique differentiable branch; implicit differentiation computes its derivative without first solving explicitly.

Put $s=\sin \theta$. By part (i), $\sin \varphi =\sqrt{2}s$. The acute-angle geometry requires $0<s<1/\sqrt{2}$. Expressing the tangents in terms of their sines in the geometric constraint gives the required implicit equationThe hint suggests $s=1/m$ for an integer $m\ge 2$. Testing $m=2$ givesThus $\boxed{\sin \theta =1/2}$. Each term on the left is strictly increasing in $s$ over the allowed interval, so this solution is unique.