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Question: Find the derivative of cosec x.


A ) cosec xcot x
B ) -cosec xcot x
C ) -sec xtan x
D ) sec xtan x

Steps to derive

1 Derivative of cosec x = ddx(cosec x)

2 = limh→0cosec (x+h)-cosec xh
 [Use the definition of derivative of function.]

3 = limh→01sin (x+h)-1sin xh

4 = limh→0sin x-sin (x+h)hsin (x+h)sin x

5 = limh→02cos (x+x+h2)sin (x-x-h2)hsin (x+h)sin x
 [Use sin A - sin B = 2cosA+B2sinA-B2.]

6 = -2limh→0cos (x+h2)sin (h2)hsin (x+h)sin x

7 = -2(limh→0cos (x+h2)sin (x+h)sin x)(limh→0sin (h2)h)
 [Use the product law of limits.]

8 = -2(cos (x+0)sin (x+0)sin x)(12)
 [Use limθ→0sin kθθ=k.]

9 = -(cos xsin xsin x)=-cosec xcot x



Hence the right answer is Option B

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