Let's break down the problem step by step to identify the missing expression in step 7.
We start with the given expression:
[tex]\[
\left(\sqrt{1+d^2}\right)^2 + \left(\sqrt{e^2+1}\right)^2
\][/tex]
First, we simplify the expression:
[tex]\[
\left(\sqrt{1+d^2}\right)^2 + \left(\sqrt{e^2+1}\right)^2 = (1 + d^2) + (e^2 + 1)
\][/tex]
This simplifies further to:
[tex]\[
1 + d^2 + e^2 + 1 = 2 + d^2 + e^2
\][/tex]
Next, compare the simplified expression with the right side of the equation in the problem statement:
[tex]\[
2 + d^2 + e^2 = d^2 - 2de + e^2
\][/tex]
Subtracting \(d^2 + e^2\) from both sides:
[tex]\[
2 = -2de
\][/tex]
Dividing both sides by \(-2\):
[tex]\[
-1 = de
\][/tex]
To understand better which expression fits into the given context, let's consider the missing relation that transforms the given expressions into a form that matches the expanded equation \(d^2 - 2de + e^2\). Notice that:
[tex]\[
d^2 - 2de + e^2 = (d - e)^2
\][/tex]
Thus, the transformation from \((1 + d^2) + (e^2 + 1)\) to \(d^2 - 2de + e^2\) implies a perfect square. Therefore, the missing expression must look like:
[tex]\[
(d - e)^2
\][/tex]
So, analyzing the options:
A. \(A^2+B^2\) does not involve \(de\) term.
B. \((A+B)^2\) expands to \(A^2 + 2AB + B^2\) - not matching our required transformation with a negative sign.
C. \((d-e)^2\) correctly expands to \(d^2 - 2de + e^2\).
D. \(-2de\) is part of the expansion but not the whole expression.
Therefore, the missing expression from step 7 is:
[tex]\[
\boxed{(d - e)^2}
\][/tex]
