If 3 angles a b c are in
a.P prove that cotb =sin-sinc
Answers
Answered by
1
Is it....????
Since a, b, and c are in A.P., there is a d such that b = a + d and c = a + 2d. Using sum-to-product identities, we have
sin a - sin c = 2 cos (a + c)/2 sin (a - c)/2 = 2 cos (a + d) sin (-d) = -2 cos b sin d,
cos c - cos a = 2 sin (a + c)/2 sin (a - c)/2 = 2 sin (a + d) sin (-d) = -2 sin b sin d.
Therefore
(sin a - sin c)/(cos c - cos a) = (-2 cos b sin d)/(-2 sin b sin d) = cos b / sin b = cot b.
Since a, b, and c are in A.P., there is a d such that b = a + d and c = a + 2d. Using sum-to-product identities, we have
sin a - sin c = 2 cos (a + c)/2 sin (a - c)/2 = 2 cos (a + d) sin (-d) = -2 cos b sin d,
cos c - cos a = 2 sin (a + c)/2 sin (a - c)/2 = 2 sin (a + d) sin (-d) = -2 sin b sin d.
Therefore
(sin a - sin c)/(cos c - cos a) = (-2 cos b sin d)/(-2 sin b sin d) = cos b / sin b = cot b.
Similar questions