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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. $\cos36^{\circ} - \cos72^{\circ}.$

fig2 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). fig 3