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If f(x) = sin x / √(1 + tan² x) - cos x / √(1 + cot² x), what is the set of all possible values of f(x)?

1. [-1,0]
2. [0,1]
3. [-1,1]
4. none of these

Correct answer: [-1,1]

Solution

Since sqrt(1+tan^2 x)=1/|cos x| and sqrt(1+cot^2 x)=1/|sin x|, f(x)=sin x|cos x| - cos x|sin x|. In quadrant II this equals -2 sin x cos x in (0,1], in quadrant IV it is in [-1,0), and 0 at the axes, so the value set is [-1, 1].

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