For the curve shown in the diagram, the normal to the curve at the point $$\(P\)$$ with coordinates $$\((x, y)\)$$ meets the $$\(x\)$$-axis at $$\(N\)$$. The point $$\(M\)$$ is the foot of the perpendicular from $$\(P\)$$ to the $$\(x\)$$-axis. The curve is such that for all values of $$\(x\)$$ in the interval $$\(0 \leqslant x<\frac{1}{2} \pi\)$$, the area of triangle $$\(P M N\)$$ is equal to $$\(\tan x\)$$. Hence show that $$\(x\)$$ and $$\(y\)$$ satisfy the differential equation $$\(\frac{1}{2} y^{2} \frac{\mathrm{d} y}{\mathrm{~d} x}=\tan x\)$$. ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................

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