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If tan(cot x)=cot(tan x), then which of the following is true?

1. sin 2x=(2)/((2n+1)π)
2. sin x=(4)/((2n+1)π)
3. sin 2x=(4)/((2n+1)π)
4. None of these

Correct answer: sin 2x=(4)/((2n+1)π)

Solution

tan(cot x) = cot(tan x) = tan(π/2 - tan x), so cot x = π/2 - tan x + nπ, giving cot x + tan x = (2n+1)π/2. Since cot x + tan x = 2/sin2x, we get sin2x = 4/((2n+1)π).

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