If in a triangle ABC, `(1 + cos A)/(a) + (1 + cos B)/(b) + (1+ cos C)/(c) = (k^(2) (1 + cos A) (1 + cos B) (1 + cos C))/(abc)`, then k is equal to

Options

A. `(1)/(2 sqrt2R)`
B. 2R
C. `(1)/(R)`
D. none of these

Answer

B

Explanation

`L.H.S. = (1 + cos A)/(a) + (1 + cos B)/(b) + (1 + cos C)/(c)`
`= (2 cos^(2).(A)/(2))/(2R sin A) + (2 cos^(2).(B)/(2))/(2R sin B) + (2 cos^(2).(C)/(2))/(2R sin C)`
`= (1)/(2R) (cot.(A)/(2) + cot.(B)/(2) + cot.(C)/(2))`
Now, `(1 + cos A)/(a) .(1 + cos B)/(b) .(1 + cos C)/(c)`
`= (cos^(2).(A)/(2))/(R sin A).(cos^(2).(B)/(2))/(R sin B) .(cos^(2)/(C)/(2))/(R sin C)`
`= (1)/(8R^(3)) cot.(A)/(2) cot.(B)/(2) cot.(C)/(2)`