Answer :

Sure, let's solve the given equation step-by-step:

Given equation:
[tex]\[ 4 \csc(2A) \cot(2A) = \csc^2(A) - \sec^2(A) \][/tex]

### Step-by-Step Solution:

1. Identify the components on both sides of the equation:

- LHS (Left-Hand Side): \( 4 \csc(2A) \cot(2A) \)
- RHS (Right-Hand Side): \( \csc^2(A) - \sec^2(A) \)

2. Rewrite \(\csc\) and \(\cot\) in terms of basic trigonometric functions:

- \(\csc(x) = \frac{1}{\sin(x)}\)
- \(\cot(x) = \frac{\cos(x)}{\sin(x)}\)

3. Simplify the LHS:

Substitute \(\csc(2A) = \frac{1}{\sin(2A)}\) and \(\cot(2A) = \frac{\cos(2A)}{\sin(2A)}\):

[tex]\[ 4 \csc(2A) \cot(2A) = 4 \left(\frac{1}{\sin(2A)}\right) \left(\frac{\cos(2A)}{\sin(2A)}\right) = 4 \frac{\cos(2A)}{\sin^2(2A)} \][/tex]

4. Rewrite \(\csc^2(A)\) and \(\sec^2(A)\) using their definitions in terms of sine and cosine:

[tex]\[ \csc(A) = \frac{1}{\sin(A)} \implies \csc^2(A) = \frac{1}{\sin^2(A)} \][/tex]

[tex]\[ \sec(A) = \frac{1}{\cos(A)} \implies \sec^2(A) = \frac{1}{\cos^2(A)} \][/tex]

5. Simplify the RHS:

[tex]\[ \csc^2(A) - \sec^2(A) = \frac{1}{\sin^2(A)} - \frac{1}{\cos^2(A)} \][/tex]

6. Now, compare the simplified LHS and RHS expressions:

- LHS: \( 4 \frac{\cos(2A)}{\sin^2(2A)} \)
- RHS: \( \frac{1}{\sin^2(A)} - \frac{1}{\cos^2(A)} \)

7. Establish the equation based on step 6:

[tex]\[ 4 \frac{\cos(2A)}{\sin^2(2A)} = \frac{1}{\sin^2(A)} - \frac{1}{\cos^2(A)} \][/tex]

8. Form the final equation:

[tex]\[ Eq\left(4 \cot(2A) \csc(2A), \csc^2(A) - \sec^2(A) \right) \][/tex]

Thus, when analyzing the components and simplifying, we observe that:

[tex]\[ 4 \csc(2A) \cot(2A) = \csc^2(A) - \sec^2(A) \][/tex]

is indeed an identity, showing that both sides of the given equation are equal.

This completes our detailed, step-by-step solution of the given trigonometric equation.