Geometry as well as trigonometry Using law of sines prove that for any triangle in which either $B$ or $C$ is obtuse 
 $$a = b\cos C + c\cos B$$
Use the law of sines to prove the additive property 
$$\sin(B+C)= \sin B\cos C + \cos B\sin C$$
These are the first questions in the book geometry revisited.
I know I have to try all the questions myself but I have tried thinking about these questions for two days, but I couldn't find convincing questions. 
I have proved the first one by using the hint given in The book. I don't t know why some fellow people put this question on hold. It involves some the concept of both trigonometry and geometry.
 A: You have copied the exercise incorrectly from the book by Coxeter and Greitzer. (I assume you are using the 1967 edition of Geometry Revisited from the New Mathematical Library, an excellent series of small books published by the MAA.)
The verbatim text of the exercise is:

  
*
  
*Show that, for any triangle $ABC,$ even if $B$ or $C$ is an obtuse angle, $a = b \cos C + c \cos B.$ Use the Law of Sines to deduce the "addition formula"
  $$ \sin (B+C) = \sin B \cos C + \sin C \cos B. $$
  

This exercise has two parts.
The first is to show that $a = b \cos C + c \cos B.$
Contrary to what you wrote,
the exercise does not say to use Law of Sines for this part.
Indeed, I do not think it makes any sense to use the Law of Sines for this.
Well-constructed figures with one or two added lines could do the job;
I'd be tempted to use Cartesian coordinates in order to avoid the
need to consider special cases. (Coordinates let you deal with
negative values of cosines much like you deal with positive values.)
Also contrary to what you wrote, the exercise does not state that
either $B$ or $C$ is obtuse; it merely says that one of them
might be obtuse, that is, you cannot consider only the cases in which
both $B$ and $C$ are acute or in which one of them is a right angle.
In fact, you must consider all possible triangles.
If assuming that $B$ is obtuse helps you prove the formula, that's fine;
that covers the case where $B$ is obtuse,
but then you must also somehow prove the case where $B$ is not obtuse.
After you show that $a = b \cos C + c \cos B,$
you are then to apply the Law of Sines in order to prove the angle addition formula for sines,
using the already-proven fact that $a = b \cos C + c \cos B.$
You will make it easier for yourself if you copy the addition formula
exactly as written in the book, rather than writing out the
formula for $\sin(A+B)$ that you have probably seen somewhere else.
The authors chose the particular symbols in their version of the formula
in order to help you more easily use the Law of Sines to derive
the addition formula from the equation $a = b \cos C + c \cos B.$
You can use the Law of Sines to make direct substitutions in that formula
that get you almost all the way to the addition formula.
If the exercise had given the formula in the
the form that you have written, the first step would be to
realize that the formula is just as valid (or not valid) for $B$ and $C$
as for $A$ and $B,$ and rename the variables so that you are proving that
$ \sin (B+C) = \sin B \cos C + \sin C \cos B. $
