Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Find the answer

Evaluate sin⁡(arcsin⁡(0.4)+arcsin⁡(0.5))⋅sin⁡(arcsin⁡(0.5)−arcsin⁡(0.4))\sin (\arcsin (0.4)+\arcsin (0.5)) \cdot \sin (\arcsin (0.5)-\arcsin (0.4)) where for x∈[−1,1]x \in[-1,1], arcsin⁡(x)\arcsin (x) denotes the unique real number y∈[−π,π]y \in[-\pi, \pi] such that sin⁡(y)=x\sin (y)=x.

A number or a short expression. Spacing and $ signs are ignored.

Solution

Use the difference of squares identity 1 to get 0.52−0.42=0.32=0.09=91000.5^{2}-0.4^{2}=0.3^{2}=0.09=\frac{9}{100}.

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