For `0 lt theta lt pi/2`, the solution (s) of
`sum_(m=1)^(6) cosec (theta+ ((m-1))/4 pi) 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

For `0 lt theta lt (pi)/(2)`
`underset(m=1)overset(6)Sigmacosec (theta+((m-1)pi)/(4))cosec(theta + (mpi)/(4))=4sqrt(2)`
`implies underset(m=1)overset(6)Sigma (1)/(sin.(theta+((m-1)pi)/(4))sin(theta+(mpi)/(4)))=4sqrt(2)`
`implies underset(m=1)overset(6)Sigma sin[theta+(mpi)/(4)-(theta+((m-1)pi)/(4))]/(sin.(pi)/(4){sin .(theta+((m-1)pi)/(4))sin.(theta+(mr)/(4))})=4sqrt(2)`
`implies underset(m=1)overset(6)Sigma(cot(theta+((m-1)pi)/(4))-cot(theta+(mpi)/(4)))/(1//sqrt(2))=4sqrt(2)`
`implies underset(m=1)overset(6)Sigma[cot(theta+((m-1)pi)/(4))-cot(theta+(mpi)/(4))]=4`
`impliescot(theta)-cot(theta+(pi)/(4))+cot(theta+(pi)/(4))-cot(theta+(2pi)/(4))+...+cot(theta+(5pi)/(4))-cot(theta+(6pi)/(4))=4`
`implies cot theta - cot((3pi)/(2)+theta)=4`
`implies cot theta + tan theta = 4`
`implies tan^(2)theta - 4 tan theta + 1 = 0`
`implies (tan theta - 2)^(2) - 3 = 0`
`implies (tan theta-2 + sqrt(3))(tan theta-2-sqrt(3))=0`
`implies tan theta = 2 -sqrt(3) or tan theta = 2 + sqrt(3)`
`implies theta = (pi)/(12), theta = (5pi)/(12) " "[because theta in (0, (pi)/(2))]`