Solve the following system of equations for x and y `5^(cosec^2x-3sec^2y)=1` and `2^(2cosecx+sqrt(3)|secy|)=64`

Answer

`x=n pi+(-1)^(n)pi//6, n in Z` and `y = m pi pm (pi)/(6), m in Z`

Explanation

`5^((cosec^(2)x-3sec^(2)y))=1=5^(@)`
`2^((2cosec x + sqrt(3)|sec y|))=64=2^(6)`
`rArr cosec^(2)x-3 sec^(2)y=0`
And `2cosec x + sqrt(3) |sec y| = 6`
Putting the value of `sqrt(3)|sec y|` ffrom equation (2) in (1)
`cosec^(2)x-(6-2 cosec)^(2)=0`
`rArr cosec^(2)x-36 +24 cosec x - 4 cosec^(2) x =0`
`rArr 3cosec^(2) x-24 cosec x=36 =0`
`rArr cosec^(2)x-8 cosec x + 12 =0`
`rArr cosec x = 2,6`
Putting the value of cosec x in equation (2), we get
`|sec y| =(2)/(sqrt(3))`
`rArr y= m pi pm (pi)/(6), m in Z`
`rArr x = n pi + (-1)^(n) pi//6, n in Z`