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Let x and y be real numbers such that sin(x)/sin(y) = 3 and cos(x)/cos(y) = 1/2. The value of sin(2x)/sin(2y) + cos(2x)/cos(2y) can be written as p/q where p and q are coprime positive integers. Find p + q.

1. 105
2. 106
3. 107
4. 108

Correct answer: 107

Solution

The first ratio simplifies immediately by the double angle identity. The second ratio requires finding sin²(y) from the constraint that sin²(x)+cos²(x)=1.

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