Let a, b, c be the sides of `Delta`ABC opposite to angles
A, b,C respecitvely.
Let `alpha = sum_(r=0)^(n) ""^(n)C_(r) b^(n-r) c^(r) cos{rB - (n-r)C}`
and `beta = sum_(r=0)^(n) ""^(n)C_(r) b^(n-r) c^(r) sin{rB - (n-r)C}`
Statement -1: `alpha = alpha^(n)`
Statement-2: `beta = alpha^(n)`

Options

A. 1
B. 2
C. 3
D. 4

Answer

c

Explanation

We have,
`alpha + i beta = sum_(r=0)^(n) ""^(n)C_(r) b^(n-r) c^(r) e^(i[rB -(n-r)C])`
`rArr alpha + i beta= sum_(r=0)^(n) ""^(n)C_(r) (be^(-ic))^(n-r) (ce^(iB))^%(r)`
`rArr alpha + i beta= (b e ^(-iC) + ce^(iB))^(n)`
`rArr alpha + i beta = {(b cos C + c cos B) + i (-b sin C + csin B)}^(n)`
`rArr alpha + i beta = (a + i0)^(n)` "" `[{:(because a = b cos C + c sin B),( " "&(b)/(sinB) = (c)/(sinC)):}]`
`rArr alpha + i beta = a^(n)`
`rArr alpha = a^(n) and beta = 0.`