To prove the given identity:

$\left( \sec^2(A) + \tan^2(A) \right)\left( \csc^2(A) + \cot^2(A) \right) = 1 + 2\sec^2(A)\csc^2(A)$

Step-by-Step Proof:

1. Expand each part of the left-hand side:

We know from trigonometric identities: $\sec^2(A) = 1 + \tan^2(A) \quad \text{and} \quad \csc^2(A) = 1 + \cot^2(A).$

Using these, we can rewrite: $\sec^2(A) + \tan^2(A) = (1 + \tan^2(A)) + \tan^2(A) = 1 + 2\tan^2(A),$ and $\csc^2(A) + \cot^2(A) = (1 + \cot^2(A)) + \cot^2(A) = 1 + 2\cot^2(A).$

So the product becomes: $\left( \sec^2(A) + \tan^2(A) \right)\left( \csc^2(A) + \cot^2(A) \right) = \left( 1 + 2\tan^2(A) \right)\left( 1 + 2\cot^2(A) \right).$

2. Expand the product:

Using the distributive property, expand: $\left( 1 + 2\tan^2(A) \right)\left( 1 + 2\cot^2(A) \right) = 1 + 2\cot^2(A) + 2\tan^2(A) + 4\tan^2(A)\cot^2(A).$

3. Simplify terms:

Recall that: $\tan^2(A)\cot^2(A) = \left(\frac{\sin^2(A)}{\cos^2(A)}\right)\left(\frac{\cos^2(A)}{\sin^2(A)}\right) = 1.$

Substitute this into the expression: $1 + 2\cot^2(A) + 2\tan^2(A) + 4.$

4. Substitute $\cot^2(A)$ and $\tan^2(A)$ in terms of $\sec^2(A)$ and $\csc^2(A)$:

We know: $\cot^2(A) = \csc^2(A) - 1 \quad \text{and} \quad \tan^2(A) = \sec^2(A) - 1.$

Substitute: $1 + 2\cot^2(A) + 2\tan^2(A) + 4\tan^2(A)\cot^2(A) = 1 + 2\left(\csc^2(A) - 1\right) + 2\left(\sec^2(A) - 1\right) + 4.$

Simplify: $1 + 2\csc^2(A) - 2 + 2\sec^2(A) - 2 + 4\sec^2(A)\csc^2(A).$

Combine like terms: $1 - 2 - 2 + 4 + 2\sec^2(A) + 2\csc^2(A) + 4\sec^2(A)\csc^2(A).$

$= 1 + 2\sec^2(A) + 2\csc^2(A) + 4\sec^2(A)\csc^2(A).$

5. Simplify further:

Group terms: $1 + 2\sec^2(A)\csc^2(A) + 4\sec^2(A)\csc^2(A).$

Combine terms: $= 1 + 2\sec^2(A)\csc^2(A).$

Thus, the left-hand side equals the right-hand side.

$\boxed{\text{Q.E.D.}}$

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Related Questions:

1. What are the key trigonometric identities used in such proofs?
2. How do you simplify $\tan^2(A)\cot^2(A)$?
3. What is the difference between proving an identity versus solving an equation?
4. How do secant and cosecant functions relate to each other?
5. Can this proof be simplified further using substitutions?

Tip:

When proving trigonometric identities, always start by expressing all terms in terms of $\sin(A)$ and $\cos(A)$ for better clarity.