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Question:

All the pairs (x,y) that satisfy the inequality 2√sin2x - 2sinx + 5.14sin2y ≤ 1 also satisfy the equation: 2sinx = siny, sinx = |siny|, sinx = 2siny, 2|sinx| = 3siny

sinx=2siny

2|sinx|=3siny

2sinx=siny

sinx=|siny|

Solution:

2√sin2x - 2sinx + 5.14sin2y ≤ 1 ⇒ 2√(sinx - 1)² + 4 ≤ 22sin2y ⇒ √(sinx - 1)² + 4 ≤ 2sin2y ⇒ sinx = 1 and |siny| = 1