To solve the given equation [tex]\((\sin A + \csc A)^2 + (\cos A + \sec A)^2 = \tan^2 A + \cot^2 A + 7\)[/tex], let's break it down step by step and verify if both sides are indeed equal.
### Step 1: Simplify the Left-Hand Side (LHS)
We start with the expression on the left-hand side of the equation:
[tex]\[
(\sin A + \csc A)^2 + (\cos A + \sec A)^2
\][/tex]
First, expand each square term:
[tex]\[
(\sin A + \csc A)^2 = \sin^2 A + 2\sin A \cdot \csc A + \csc^2 A
\][/tex]
[tex]\[
(\cos A + \sec A)^2 = \cos^2 A + 2\cos A \cdot \sec A + \sec^2 A
\][/tex]
Since [tex]\(\csc A = \frac{1}{\sin A}\)[/tex] and [tex]\(\sec A = \frac{1}{\cos A}\)[/tex], we have:
[tex]\[
\sin A \cdot \csc A = \sin A \cdot \frac{1}{\sin A} = 1
\][/tex]
[tex]\[
\cos A \cdot \sec A = \cos A \cdot \frac{1}{\cos A} = 1
\][/tex]
Substituting these into the expanded form gives:
[tex]\[
(\sin^2 A + 2 \cdot 1 + \frac{1}{\sin^2 A}) + (\cos^2 A + 2 \cdot 1 + \frac{1}{\cos^2 A})
\][/tex]
Combine the terms:
[tex]\[
\sin^2 A + \csc^2 A + 2 + \cos^2 A + \sec^2 A + 2
\][/tex]
Since [tex]\(\csc^2 A = 1 + \cot^2 A\)[/tex] and [tex]\(\sec^2 A = 1 + \tan^2 A\)[/tex], substitute these identities:
[tex]\[
\sin^2 A + (1 + \cot^2 A) + 2 + \cos^2 A + (1 + \tan^2 A) + 2
\][/tex]
Simplify the expression:
[tex]\[
\sin^2 A + \cos^2 A + \cot^2 A + \tan^2 A + 1 + 1 + 2 + 2
\][/tex]
Using the Pythagorean identity [tex]\(\sin^2 A + \cos^2 A = 1\)[/tex], we get:
[tex]\[
1 + \cot^2 A + \tan^2 A + 6
\][/tex]
So, the simplified form of the left-hand side is:
[tex]\[
\tan^2 A + \cot^2 A + 7
\][/tex]
### Step 2: Simplify the Right-Hand Side (RHS)
We take the expression on the right-hand side of the equation, which is already simplified:
[tex]\[
\tan^2 A + \cot^2 A + 7
\][/tex]
### Step 3: Compare LHS and RHS
Notice that both sides of the equation are already in the same simplified form:
[tex]\[
\tan^2 A + \cot^2 A + 7
\][/tex]
### Conclusion
Since the left-hand side simplifies to exactly the same form as the right-hand side, we can conclude that the given equation holds true.
Therefore:
[tex]\[
(\sin A + \csc A)^2 + (\cos A + \sec A)^2 = \tan^2 A + \cot^2 A + 7
\][/tex]
Thus, the equation is correct and verified.
