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If x sec theta , y = tan theta , then...

If ` x sec theta , y = tan theta ` , then the value of ` (d^(2) y)/(dx^(2)) " at " theta = (pi)/(4)` is

A

0

B

1

C

`-1`

D

2

Text Solution

Verified by Experts

The correct Answer is:
C
Given , ` x = sec theta " and " y = tan theta `
`rArr (dx)/(d theta ) = sec theta tan theta " and " (dy)/(d theta ) = sec^(2) theta `
` therefore (dy)/(dx) = (sec^(2) theta )/(sec theta tan theta) = "cosec " theta `
Now , `(d^(2) y)/(dx^(2)) = (d)/(dx) ((dy)/(dx))`
` = (d)/(d theta)("cosec " theta )(d theta )/(dx) `
`= 0 "cosec "theta cot theta xx (1)/(sec theta tan theta ) = - (1) /(tan^(3) theta)`
`(d^(2) y)/(dx^(2)) = (d)/(dx) ((dy)/(dx))`
` = (d)/(d theta)("cosec " theta )(d theta )/(dx) `
`= 0 "cosec "theta cot theta xx (1)/(sec theta tan theta ) = - (1) /(tan^(3) theta)`
`((d^(2)y)/(dx^(2)))""_(theta = (pi)/(4))= - (1)/((tan""(pi)/(4))^(3) ) = - 1`
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