In `Delta ABC`, which one is true among the following ?

Options

A. `(b+c)"cos"(A)/(2)=sin((B+C)/(2))`
B. `(b+c)"cos"((B+C)/(2))=alpha"sin"(A)/(2)`
C. `(b-c)"cos"((B-C)/(2))=alpha"cos"((A)/(2))`
D. `(b-c)"cos"(A)/(2)=sin((B-C)/(2))`

Answer

D

Explanation

Let a,b,c are the sides of `DeltaABC`,
Now, `(b+c)/(c)=(k (sin B+sin C))/(k sin A)` [ by sine rule]
`=(2sin ((B+C)/(2))cos((B-C)/(2)))/(2"sin"(A)/(2)"cos"(A)/(2))rArr(b+c)/(a)=(cos ((B-C)/(2)))/("sin"(A)/(2))`
Also, `(b-c)/(a)=(sin ((B-C)/(2)))/("cos"(A)/(2))`