If `x=e^(theta)(sintheta-costheta),y=e^(theta)(sintheta+costheta)`, then `(dy)/(dx)` at `theta=(pi)/(4)` is

Options

A. 1
B. 0
C. `(1)/(sqrt(2))`
D. `sqrt(2)`

Answer

A

Explanation

Given ,
`x = e^(theta)(sin theta- cos theta)`
and `y = e^(theta) ( sin theta + cos theta)`
`rArr (dx)/(d theta)= e^(theta) ( sin theta - cos theta + cos theta + sin theta)`
`= 2 e^(theta) sin theta`
and `(dy)/(d theta) = e^(theta) ( sin theta + cos theta + cos theta - sin theta)`
`= 2e^(theta) cos theta`
`:.(dy)/(dx)=((dy)/(d theta))/((dx)/(d theta))=(2e^(theta) cos theta)/( 2 e ^(theta) sin theta ) = cot theta`
`rArr ((dy)/(dx))_(theta=pi/4)= cot . (pi)/(4)=1`