`(cos^2 x+1/(cos^2 x))(1+tan^2 2 y)(3+sin 3 z)=4,` then y can take values equal to

Options

A. x is an integral multiple of `pi`
B. x cannot be an even multiple of `pi`
C. z is an integral multiple of `pi`
D. y is an integral multiple of `pi//2`

Answer

A, D

Explanation

`(cos^(2) x+1/(cos^(2) x)) (1+tan^(2) 2y) (3+ sin 3z)=4`
Since `cos^(2) x+1/(cos^(2) x) ge 2, 1 + tan^(2) 2y ge 1`
and `2 le 3+ sin 3z le 4`, the only possibility is
`cos^(2) x+1/(cos^(2) x)=2, 1 + tan^(2) 2y=1, 3+sin 3z=2`
Thus, `cos x= pm 1 rArr x=npi`,
`tan 2y=0rArr y=(m pi)/2` and `sin 3z=-1`
`rArr z=(4k-1) pi/6, m, n, k in I`