If in a triangle `A B C ,(1+cosA)/a+(1+cosB)/b+(1+cosC)/c` `=(k^2(1+cosA)(1+cosB)(1+cosC)/(a b c)` , then `k` is equal to (a) `1/(2sqrt(2)R)` (b) `2R` (c) `1/R` (d) none of these

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)`