If `(1+sqrt(1+x))tanx=1+sqrt(1-x),` then sin 4x is equal to

Options

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

Answer

C

Explanation

We have,
`tanx=(1+sqrt(1-x))/(1+sqrt(1+x))`
`impliestanx=(1+sqrt(1-sintheta))/(1+sqrt(1+sintheta)),where x=sintheta`
`impliestanx=(1+cos""(theta)/(2)-sin""(theta)/(2))/(1+cos^(2)""(theta)/(2)+sin""(theta)/(2))`
`impliestanx=(1+cos""(theta)/(4)-2sin""(theta)/(4)cos""(theta)/(4))/(2cos^(2)""(theta)/(4)+2 sin""(theta)/(4)cos""(theta)/(4))`
`impliestanx=(1-tan""(theta)/(4))/(1+tan""(theta)/(4))`
`impliestanx =tan((pi)/(4)-(theta)/(4))`
`impliesx=(pi)/(4)=(theta)/(4)`
`implies4x=pi-thetaimpliessin4x=sin(pi-theta)=sintheta=x`