For `0 lt theta lt pi/2`, the solutions of `sigma_(m-1)^(6)"cosec"(theta+((m-1)pi)/4)" cosec "(theta+(mpi)/4)=4sqrt(2)` is (are):

Options

A. `pi//4`
B. `pi//6`
C. `pi//12`
D. `5pi//12`

Answer

C, D

Explanation

`sum_(m=1)^(6) cosec (theta +((m-1))/4 pi) cosex (theta +(mpi)/4)=4sqrt(2)`
`rArr 1/(sin (pi//4))[(sin (theta+pi//4-theta))/(sin theta. sin (theta+pi//4))+(sin ((theta+pi//2)-(theta+pi//4)))/(sin (theta+pi//4).sin (theta+pi//2))+..+(sin ((theta+3pi//2)-(theta+5pi//4)))/(sin (theta+3pi//2). sin (theta+5pi//4))]=4sqrt(2)`
`rArr 1/(sin (pi//4))[(sin (theta+pi//4) cos theta-cos (theta+pi//4) sin theta)/(sin theta. sin (theta+pi//4))+(sin (theta+pi//2) cos (theta+pi//4)-cos (theta+pi//2) sin (theta+pi//4))/(sin (theta+pi//4). sin (theta+pi//2))+... (sin (theta+3pi//2)cos (theta+5pi//4)-cos (theta+3pi//2) sin (theta+5pi//4))/(sin (theta+3pi//2). sin (theta+5pi//4))]=4sqrt(2)`
`rArr sqrt(2) [cot theta-cot (theta+pi//4)+cot(theta+pi//4)-cot (theta+pi//2)+...+cot (theta+5pi//4)-cot(theta+3pi//2)]=4sqrt(2)`
`rArr tan theta+cot theta=4`
`rArr tan theta=2 pm sqrt(3)`
`rArr theta=pi/12 or (5pi)/12`