The system of equations `tan x=a cot x, tan 2x=b cos y`

Options

A. cannot have a solution if `a=0`
B. cannot have a solution if `a=1`
C. cannot have a solution if `2sqrt(a) gt |b(1-a)|`
D. has a solution for all a and b

Answer

B, C

Explanation

If `a=0` then `tanx=0rArr x= npi` and for any value of y such that `cos y =0` the second equation is satisfied.
So, option (1) is not correct.
If `a=1` then `tan x=cot x rArr tan^(2) x=1` i.e., `tan 2x` is not defined.
So, option (2) is correct.
Now, from the first equation, `tan x=sqrt(a)` i.e., a must be positive.
`rArr` Now, `|cos y|=|(2 tan x)/(b(1-tan^(2) x))|=|(2sqrt(a))/(b(1-a))| le 1`
`rArr 2sqrt(a) le |b (1-a)|`