If `A,B,C` are the angles of a triangle, the system of equations
`(sinA)x+y+z=cosA`, `x+(sinB)y+z=cosB`, `x+y+(sinC)z=1-cosC` has

Options

A. No solution
B. Unique solution
C. Infinitely many solutions
D. Finitely many solutions

Answer

B

Explanation

`(b)` Let `Delta=|{:(sinA,1,1),(1,sinB,1),(1,1,sinC):}|`
Apply `C_(2)toC_(2)-C_(1)` and `C_(3)toC_(3)-C_(1)`
Then expand along `C_(1)`, we get
`Delta=sinA(1-sinB)(1-sinC)+(1-sinA)(1-sinB)+(1-sinA)(1-sinC)`
Since `A,B,C` are angles of a triangle, `0 lt sin A, sinB,sinC le 1`
`implies Delta ne 0`