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If e^(x)=(sqrt(1+z)-sqrt(1-z))/(sqrt(1+...

If `e^(x)=(sqrt(1+z)-sqrt(1-z))/(sqrt(1+z)+sqrt(1-z))` and `tan (y/2)=sqrt((1-z)/(1+z))` then the value of `(dy)/(dx)` at `z=1` is equal to to

A

`-2`

B

`-1`

C

0

D

1

Text Solution

Verified by Experts

The correct Answer is:
A
`"Using"z=costheta," "atz=1rArrtheta=0^(@)`
Now, `e^(x)=((costheta)/(2)-"sin"(theta)/(2))/((costheta)/(2)+"sin"(theta)/(2))=((pi)/(4)-(theta)/(2))`
`:.x=lntan((pi)/(4)-(theta)/(2))`
`(dx)/(d theta)=(1)/("tan"((pi)/(4)-(theta)/(2)))sec^(2)((pi)/(4)-(theta)/(2)).((-1)/(2))rArr(dx)/(d theta)=-sectheta`
Also, `"tan"(y)/(2)="tan"(theta)/(2)rArr(y)/(2)=tan^(-1)("tan"(theta)/(2))rArr(y)/(2)=(theta)/(2)rArr(dy)/(d theta)=1`
Hence `(dy)/(dx)=-costheta" at "theta=0^(@)rArr(dy)/(dx)=-1`.
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