For `0ltthetalt pi/ 2`,the solution of`sum_(m=1)^6cosec(theta+((m-1)pi)/4)cosec (theta +( mpi)/4)=4sqrt2` is(are);

Options

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

Answer

C, D

Explanation

We have,
`overset(6)underset(m=1)(sum)"cosec" (theta + (m-1)(pi)/(4))"cosec" (theta + (mpi)/(4)) = 4sqrt(2)`
`rArr sum_(m=1)^(6) ("sin"{(theta + (mpi)/(4)) - (theta + (m-1)(pi)/(4))})/("sin"(theta + (mpi)/(4))"sin" (theta + (m-1)(pi)/(4))) = 4`
`rArr sum_(m=1)^(6) {"cot"(theta + (m-1)(pi)/(4)) - "cot"(theta + (mpi)/(4))}=4`
`rArr "cot"theta - "cot" ((3pi)/(2) + theta) = 4`
`rArr "cot"theta + "tan" theta = 4`
`rArr 1 +"tan"^(2) = 4 "tan" theta`
`rArr "tan"^(2) theta - 4"tan" theta + 1 = 0`
`"tan" theta = 2+- sqrt(3) rArr theta = (pi)/(12) "or" (5pi)/(12)`