Consider the equation `sectheta+ cottheta = 31/12`
On the basis of above, answer the following
Number of values of theta where `theta in [0,5pi]` is equal to

Options

A. `4`
B. `6`
C. `8`
D. `10`

Answer

D

Explanation

Let `t =tan,(theta)/(2)`
`implies (1+t^(2))/(1-t^(2))+(1-t^(2))/(2t)=(31)/(12)`
`implies6t^(4)+43t^(3)-12t^(2)-19t+6=0`
`implies(3t-1)(2t^(3)+15t^(2)+t-6)=0`
`implies t=(1)/(3),alpha,beta,gamma`
where `alpha in(-8,-7)implies(theta)/(2)in((pi)/(2),(3pi)/(4))`
`beta in (-1,(-1)/(2))implies(theta)/(2) in ((3pi)/(4),(7pi)/(8))`
`gamma in((1)/(2),1)implies(theta)/(2)in(0,(pi)/(4))`
(i) if `0lttheta lt (pi)/(2)` then `t=(1)/(3)` or `t in ((1)/(2),1)`
`implies tan theta =(3)/(4) "or if" tan. (theta)/(2)in((1)/(2),1)implies "tan" theta in ((4)/(3),oo)`
`implies"Min value of" [tan theta] =0`
(ii)No. of values of `theta in ((pi)/(2),(3pi)/(2))` will be 1