Statement I If `2 sin (theta/2)=sqrt(1+sin theta)+sqrt(1-sin theta)` then `theta/2` lies between `2 n pi+pi/4` and `2 n pi + (3pi)/(4)`.
Statement II If `(pi)/(4) le 0 le (3pi)/(4)` then `sin. (theta)/(2) gt0`.

Options

A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement -1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

B

Explanation

We have,
`2sin""(theta)/(2)=sqrt(1+sintheta)+sqrt(1-sintheta)`
`implies2sin""(theta)/(2)=sqrt((cos""(theta)/(2)+sin""(theta)/(2))^(2))+sqrt((cos""(theta)/(2)-sin""(theta)/(2))^(2))`
`implies2sin""(theta)/(2)=|cos""(theta)/(2)+sin""(theta)/(2)|+|cos""(theta)/(2)-sin""(theta)/(2)|`
`rArr "cos"(theta)/(2) + "sin"(theta)/(2)gt0 and "cos"(theta)/(2)-"sin"(theta)/(2) lt0`
`rArr sin ((pi)/(4)+(theta)/(2)) gt0 and cos ((theta)/(2)+(pi)/(4))lt0`
`rArr 2n pi +(pi)/(2)lt(0)/(2)+(pi)/(4)lt2n pi+pi`
`implies(8n+1)(pi)/(2)lt thetalt(8n+3)(pi)/(2)`
So, statement-1 is true.
Statement-2 is also true, but it is not a correct explanation for
Statement-1.