In the previous exercise (Section 13.14, Exercise #9) we proved the “cab minus bac” formula:
Using this formula prove the following identities:
-
.
-
.
-
if and only if
.
-
.
- Proof. Using the cab minus bac formula with
in place of
,
in place of
, and
in place of
we have
- Proof. Applying the cab minus back formula to each of the three terms in the sum we have
So, putting these together we have
- Proof. From cab minus bac we have
Furthermore, since
, we can apply bac minus cab to get
Therefore,
- Proof. From a previous exercise (Section 13.14, Exercise #7(d)) we know the identity
. In this case we have
in place of
,
in place of
and
in place of
. This gives us