To prove the given identity:

$(\sin A + \csc A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$

Step 1: Expand the left-hand side (LHS)

We start by expanding each squared term on the left-hand side.

1. Expanding $(\sin A + \csc A)^2$:

$(\sin A + \csc A)^2 = \sin^2 A + 2\sin A \cdot \csc A + \csc^2 A$ Since $\csc A = \frac{1}{\sin A}$, we have: $\sin A \cdot \csc A = 1$ Thus: $(\sin A + \csc A)^2 = \sin^2 A + 2 + \csc^2 A$ Using $\csc^2 A = \frac{1}{\sin^2 A}$, we get: $(\sin A + \csc A)^2 = \sin^2 A + 2 + \frac{1}{\sin^2 A}$

2. Expanding $(\cos A + \sec A)^2$:

$(\cos A + \sec A)^2 = \cos^2 A + 2\cos A \cdot \sec A + \sec^2 A$ Since $\sec A = \frac{1}{\cos A}$, we have: $\cos A \cdot \sec A = 1$ Thus: $(\cos A + \sec A)^2 = \cos^2 A + 2 + \sec^2 A$ Using $\sec^2 A = \frac{1}{\cos^2 A}$, we get: $(\cos A + \sec A)^2 = \cos^2 A + 2 + \frac{1}{\cos^2 A}$

Step 2: Add the expanded terms

Now, sum up both parts of the left-hand side:

$(\sin A + \csc A)^2 + (\cos A + \sec A)^2 = \left( \sin^2 A + 2 + \frac{1}{\sin^2 A} \right) + \left( \cos^2 A + 2 + \frac{1}{\cos^2 A} \right)$

Simplifying this expression: $= \sin^2 A + \cos^2 A + 4 + \frac{1}{\sin^2 A} + \frac{1}{\cos^2 A}$ Since $\sin^2 A + \cos^2 A = 1$, we have: $1 + 4 + \frac{1}{\sin^2 A} + \frac{1}{\cos^2 A} = 5 + \frac{1}{\sin^2 A} + \frac{1}{\cos^2 A}$

Step 3: Express $\frac{1}{\sin^2 A} + \frac{1}{\cos^2 A}$ in terms of $\tan A$ and $\cot A$

We know that: $\frac{1}{\sin^2 A} = \csc^2 A = 1 + \cot^2 A$ and $\frac{1}{\cos^2 A} = \sec^2 A = 1 + \tan^2 A$ Thus: $\frac{1}{\sin^2 A} + \frac{1}{\cos^2 A} = (1 + \cot^2 A) + (1 + \tan^2 A) = 2 + \cot^2 A + \tan^2 A$

Step 4: Final simplification

Substituting this into our previous expression: $5 + \frac{1}{\sin^2 A} + \frac{1}{\cos^2 A} = 5 + 2 + \cot^2 A + \tan^2 A$ $= 7 + \cot^2 A + \tan^2 A$

Thus, the left-hand side simplifies to:

$(\sin A + \csc A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$

Conclusion:

We have shown that: $(\sin A + \csc A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$ The identity is proven to be true.

---

Would you like a more detailed explanation of any specific step or have any other questions?

Here are some related questions you might find useful:

1. Can you prove the Pythagorean identity $\sin^2 A + \cos^2 A = 1$?
2. How is $\tan A$ related to $\sin A$ and $\cos A$?
3. What are the properties of $\sec A$ and $\csc A$ in trigonometry?
4. Can you derive the double angle identities for sine and cosine?
5. How does the identity $\tan^2 A + 1 = \sec^2 A$ relate to this proof?

Tip: When working with trigonometric identities, remember that many can be transformed into simpler forms using basic Pythagorean identities or reciprocal relationships between the trigonometric functions.