Statement -1: If `(pi)/(12)lethetale(pi)/(3),` then
`sin(theta-(pi)/(4))sin(theta-(7pi)/(12))sin(theta+(pi)/(12))"lies between"-(1)/(4sqrt2)and 1/4.`
Statement-2: The value of `sin thetasin((pi)/(3)-theta)sin((pi)/(3)+theta)is 1/4sin3theta.`

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

A

Explanation

We have,
`sinthetasin((pi)/(3)-theta)sin((pi)/(3)+theta)=sintheta(sin^(2)""(pi)/(3)-sin^(2)theta)`
`=sin^(2)theta((3)/(4)-sin^(2)theta)=1/4sin3theta`
So, statement-2 is true.
On replacing `thetaby(theta-(pi)/(4))` in statement-2, we get
`sin(theta-(pi)/(4))sin((7pi)/(12)-theta)sin(theta+(pi)/(12))=1/4sin(3theta-(3pi)/(4))`
`impliessin(theta-(pi)/(4))sin(theta-(7pi)/(12))sin(theta+(pi)/(12))=-1/4sin(3theta-(3pi)/(4))`
If `(pi)/(12)lethetale(pi)/(3),then (pi)/(4)le3thetalepi`
`implies-(pi)/(2)le3theta-(3pi)/(4)le(pi)/(4)`
`implies-1lesin(3theta-(3pi)/(4))le(1)/(sqrt2)`
`implies-(1)/(4sqrt2)lesin(theta-(pi)/(4))sin(theta-(7pi)/(12))sin(theta+(pi)/(12))le1/4`
So, statement-1 is also true and statement-2 is a correct explanation for statement-1.