Given that `(1+sqrt(1+x))tany=1+sqrt(1-x)` . Then `sin4y` is equal to `4x` (b) `2x` (c) `x` (d) none of these

Options

A. 4x
B. 2x
C. x
D. none of these.

Answer

C

Explanation

`tan y=(1+sqrt(1-x))/(1+sqrt(1+x))`
Let `x=cos theta,` then
`sqrt(1-x)=sqrt(2)sin(theta//2),sqrt(1+x)=sqrt(2)cos(theta//2)`
`rArr tany=(sqrt(2)[(1)/sqrt(2)+sin((theta)/(2))])/(sqrt(2)[(1)/sqrt(2)+cos((theta)/(2))])``=(sin((pi)/(4))+sin((theta)/(2)))/(cos((pi)/(4))+cos((theta)/(2)))`
`=(2sin((pi)/(8)+(theta)/(4))cos((pi)/(8)-(theta)/(4)))/(2cos((pi)/(8)+(theta)/(2))cos((pi)/(8)-(theta)/(4)))`
`=tan((pi)/(8)+(theta)/(4))`