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Gold Again

Age 16 to 18 Challenge Level: Consider the rhombus as illustrated where $x$ is an unknown length, $AP = AD = x$ , angle $DAP = 36$ degrees and $P$ is a point on the diagonal $AC$ such that $PB = 1$ unit. Without using a calculator, computer or tables find the exact values of 1. $\cos36^{\circ}\cos72^{\circ}$ 2. $\cos36^{\circ} - \cos72^{\circ}.$ 3. Draw these two diagrams as accurately as you can and measure the lengths $a$ and $b$. What do you notice? Can you prove it? (In each diagram there are two right angled triangles). 