Solve for x and y `12sinx-2y^2=21-8y-5 cosx`

Answer

`x=2npi + cos^(-1) (5/13), n in Z and y=2`

Explanation

`12 sin x +5 cos x=2y^(2) - 8y+21`
`rArr sqrt(12^(2)+5^(2))(12/13 sin x + 5/13 cos x)=2(y^(2)-4y+4)+13`
`rArr 13 cos(x-alpha)=2 (y-2)^(2)+13`
where `cos alpha = 5//13`.
Now `L.H.S. le 13 and R.H.S. ge 13`
`rArr 13 cos(x-alpha)=2 (y-2)^(2)+13=13`
`rArr cos (x-alpha)=1 and y=2`
`rArr x-alpha=2n pi and y=2`
`rArr x=2npi + cos^(-1) (5/13) and y=2`, where `n in Z`