`" if " |{:(sin x,,sin y,,sin z),(cos x,,cos y,,cos z),(cos^(3) x,,cos^(3)y,,cos^(3)z):}|=0` then which of the following is`//`are possible ?

Options

A. `x=y`
B. `y=z`
C. `x=z`
D. `x+y+z=pi//2`

Answer

A, B, C, D

Explanation

`|{:(sin x ,,sin y,,sin z),(cos x,,cos y,,cos z),(cos^(3)x,,cos^(3)y,,cos^(3)z):}|`
Taking cos x cos y and z common from `C_(1) ,C_(2) " and " C_(3)` respectively.
`Delta = " cos x cos y cos z " |{:(tan x ,,tan y,,tan z),(1,,1,,1),(cos^(2),,cos^(2) y,,cos^(2)z):}|`
Applyin `C_(1) to C_(1)-C_(2),C_(2) to C_(2) -C_(3)` we get
`Delta = " cos x cos y cos z " |{:(tan x - tan y,,tan y - tan x,,tan z),(0,,0,,1),(cos^(2)x-cos^(2)y,,cos^(2)y-cos^(2)z,,cos^(2)z):}|`
`=-" cos x cos y cos z "|{:(tan x-tan y,,tan y- tan z),(cos^(2) x-cos^(2) y,,cos^(2) y -cos^(2) z):}|`
`=- cos x cos y cos z`
`|{:((sin(x-y))/(cos x cos y),,(sin(y-z))/(cos ycos x)),(-sin (x-y) sin (x+y),,-sin (y-z) sin (y+z)):}|`
`=sin (x-y) sin (y-z) |{:(cos z,,cos x ),(sin (x+y),,sin (y+z)):}|`
`=sin (x-y) sin(y-z) [cos z sin (y+z)- cos x sin (x+y)]`
`=sin (x-y) sin (y-z) [sin y cos^(2) z+sin z cos y cos z- sin x cos y cos x -sin y cos^(2) x]`
`=sin (x-y) sin (y-z) [sin y(cos^(2) z-cos^(2)x) +cos y(sin zcos z-sin x cos x)]`
`=sin (x-y) sin (y-z)[sin y sin (x-z) sin (x-z) +cos y (sin 2z-sin 2x)//2]`
`=sin (x-y) sin (y-z)[sin y sin (x+z) sin (x-z) + cos y sin (z-x) cos (z+x)]`
`=sin (x-y) sin (y-z) sin (z-x) -sin y sin (x+z) + cos y cos (z+x)]`
`=sin (x-y) sin (y-z) sin (z-x) cos (x+y+z)`
For `Delta =0, x=y " or " y=z " or " x+y+z=pi//2`