If `cos^(-1)(x)+cos^(-1)(y)+cos^(-1)(z)=pi(sec^2(u)+sec^4(v)+sec^6(w)),w h e r e u , v , w` are least non-negative angles such that `u

Answer

9

Explanation

`sec^(2)u, sec^(4)v, sec^(6)w in [1, oo)`
`:. Sec^(2) (u) + sec^(4)(v) + sec^(6) (w) in [3, oo)`
`:. Pi (sec^(2) u + sec^(4)v + sec^(6)w) in [3pi, oo)`
But `cos^(-1) x + cos^(-1) y + cos^(-1) z in [0, 3pi]`
So equation is possible of L.H.S. = R.H.S. `= 3pi`
`:. cos^(-1) x = cos^(-1) y = cos^(-1) z = pi`
`:. x = y = z =-1`
and `sec^(2) u = sec^(4) v = sec^(6) w = 1`
`:. u = pi, v = 2pi, w = 3pi`
`:. x^(2000) + y^(2002) + z^(2004) + (36pi)/(u + v + w) = 1 + 1 + 1 + (36pi)/(6pi) = 9`