<s>[INST] Showing $\cos A\cos B\cos C=\frac{s^2-(2R+r)^2}{4R^2}$ and $\cos A+\cos B+\cos C=1+\frac rR$ in $\triangle ABC$ 
In a triangle with vertices $A$, $B$, $C$, semiperimeter $s$, inradius $r$ and circumradius $R$, prove that $$\cos A\cos B\cos C=\frac{s^2-(2R+r)^2}{4R^2}$$ and $$\cos A+\cos B+\cos C=1+\frac rR$$

(note: we can also discover the value of $\cos A\cos B+\cos B\cos C+\cos C\cos A$ using the identity $\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C=1$)
Since the last time I've posted this question (the original thread is now deleted), I've reflected a bit on the suggestions of several users. First, I included relevant informations and defintion and second I did try to use the cosine law, but It did not give me help.
I was referred by a friend to the identities
$$\begin{align}
a+b+c &= 2s \tag{1} \\[4pt]
ab+ac+bc &= s^2+r^2+4rR \tag{2} \\[4pt]
abc &= 4Rrs \tag{3}
\end{align}$$ The first and third facts are obvious, while the second I do not know for sure to be true (although it probably is) and appears to model the numerator of the first identity in $\cos A\cos B\cos C$.
Any other idea?
 [/INST] The area of the triangle can be expressed as $ \frac12r(a+b+c)= \frac{abc}{4R}$. Then
\begin{align}
\frac rR =\frac12 \frac{\frac{abc}{R^3}}{\frac{a+b+c}R}
=\frac{2\sin A \sin B \sin C}{\sin A +\sin B +\sin C}\tag1
\end{align}
Note
$$\sin A +\sin B +\sin C
= 2\cos\frac A2\sin\frac A2 + 2\sin\frac {B+C}2\cos\frac{B-C}2\\
= 2 \cos\frac A2 (\cos\frac {B+C}2+\cos\frac{B-C}2) = 4 \cos\frac A2 \cos\frac B2 \cos\frac C2
$$
Substitute into (1)
$$\frac rR =  4 \sin\frac A2 \sin\frac B2 \sin\frac C2\tag2
$$
Similarly
$$\cos A +\cos B +\cos C
= 1-2\sin^2\frac A2 +2\cos\frac {B+C}2\cos\frac{B-C}2\\
=1-  2 \sin\frac A2 (\cos\frac {B+C}2-\cos\frac{B-C}2)=1+ 4 \sin\frac A2 \sin\frac B2 \sin\frac C2
$$
Substitute into (2)
$$\cos A+\cos B+\cos C=1+\frac rR$$
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