Duality of cost minimization and profit maximization
If $F(K,L)$ is a homogeneous function of degree one then so is
$$
\Pi(K,L) = F(K,L) - R \cdot K - w \cdot L.
$$
This follows straight from the definition of homogeneity. (A definition of homogeneous function can be found here.) This means that if a maximal profit exists it is zero. Otherwise you could increase all inputs by say 100%, thereby increasing both revenues and costs and thus profits by 100%. So $\Pi(K^*,L^*) = 0$.

By Euler's Homogeneous Function Theorem we have $\forall K,L$:

\begin{align}
\Pi(K,L) &amp;= \Pi_K(K,L) \cdot K + \Pi_L(K,L) \cdot L  \\
\Pi(K,L) &amp;= (F_K(K,L) - R) \cdot K + (F_L(K,L) - w) \cdot L.
\end{align}

Since $\Pi(K^*,L^*) = 0$, we have
$$
-(F_K(K^*,L^*) - R) \cdot K^* = (F_L(K^*,L^*) - w) \cdot L^*
$$
We know that $K^*,L^* \geq 0$, so if we can show that the signs of 
$(F_K(K^*,L^*) - R)$ and $(F_L(K^*,L^*) - w)$ match we will have proven them to be equal to zero. Otherwise one side of the equation would be negative and the other positive. From cost minimization you have
\begin{align}
R + \lambda F_K &amp; = 0\\
w + \lambda F_L &amp; = 0.
\end{align}
If $\lambda &gt;1$ then
\begin{align}
F_K(K^*,L^*) - R &amp; &lt; 0\\
F_L(K^*,L^*) - w &amp; &lt; 0,
\end{align}
if $\lambda =1$ then
\begin{align}
F_K(K^*,L^*) - R &amp; = 0\\
F_L(K^*,L^*) - w &amp; = 0.
\end{align}
and if $\lambda &lt;1$ then
\begin{align}
F_K(K^*,L^*) - R &amp; &gt; 0\\
F_L(K^*,L^*) - w &amp; &gt; 0,
\end{align}

so the signs do indeed match, hence
\begin{align}
F_K(K^*,L^*) - R &amp; = 0\\
F_L(K^*,L^*) - w &amp; = 0.
\end{align}