First lets start with a right-angled triangle where one of the angles is y. Lets say the opposite side is of length 1 and that the adjacent is of length x:
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Using the triangle above, we can say:
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We already know that the derivative of x with respect to x is 1:
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Since x = cotan(y):
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Using the chain rule:
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And we already know what cosec(y) is equal to:
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