 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/4/a/solution

On any interval where

sin x > 0

, the chain rule gives

y^{'} = cos x / sin x = cot x

. Differentiating again,

y^{''} = - csc^{2} x = - \frac{1}{sin ^{2} x} .

The domain condition is necessary because the real natural logarithm requires a positive argument.

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