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HMMT February

Algebra Difficulty 4.7 AIME Find the answer

Given that sin⁡A+sin⁡B=1\sin A+\sin B=1 and cos⁡A+cos⁡B=3/2\cos A+\cos B=3 / 2, what is the value of cos⁡(A−B)\cos (A-B)?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Squaring both equations and add them together, one obtains 1+9/4=2+2(cos⁡(A)cos⁡(B)+sin⁡(A)sin⁡(B))=2+2cos⁡(A−B)1+9 / 4=2+2(\cos (A) \cos (B)+\sin (A) \sin (B))=2+2 \cos (A-B). Thus cos⁡A−B=5/8\cos A-B=5 / 8.

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