prove 2/cos^2A − 1/cos^4A − 2/sin^2A + 1/sin^4A = cot^4A - tan^4A with explanation

We are asked to prove the following trigonometric identity:

\[
\frac{2}{\cos^2A} - \frac{1}{\cos^4A} - \frac{2}{\sin^2A} + \frac{1}{\sin^4A} = \cot^4A - \tan^4A
\]

### Step 1: Express everything in terms of \(\sin A\) and \(\cos A\)

We know that:
- \(\cot A = \frac{\cos A}{\sin A}\)
- \(\tan A = \frac{\sin A}{\cos A}\)

The right-hand side of the equation can be expressed as:
\[
\cot^4A - \tan^4A = \left( \frac{\cos^4 A}{\sin^4 A} \right) - \left( \frac{\sin^4 A}{\cos^4 A} \right)
\]

### Step 2: Simplify the right-hand side

Using the difference of squares identity, \(a^4 - b^4 = (a^2 - b^2)(a^2 + b^2)\), we can rewrite:
\[
\cot^4A - \tan^4A = \left( \frac{\cos^2A}{\sin^2A} - \frac{\sin^2A}{\cos^2A} \right) \cdot \left( \frac{\cos^2A}{\sin^2A} + \frac{\sin^2A}{\cos^2A} \right)
\]
Now, simplify each factor:
- The first factor becomes:
  \[
  \frac{\cos^2A}{\sin^2A} - \frac{\sin^2A}{\cos^2A} = \frac{\cos^4A - \sin^4A}{\sin^2A \cos^2A}
  \]
- The second factor simplifies to:
  \[
  \frac{\cos^2A}{\sin^2A} + \frac{\sin^2A}{\cos^2A} = \frac{\cos^4A + \sin^4A}{\sin^2A \cos^2A}
  \]

Therefore, the right-hand side becomes:
\[
\frac{\cos^4A - \sin^4A}{\sin^2A \cos^2A}
\]

### Step 3: Simplify the left-hand side

Start with the original expression on the left-hand side:
\[
\frac{2}{\cos^2A} - \frac{1}{\cos^4A} - \frac{2}{\sin^2A} + \frac{1}{\sin^4A}
\]

We can combine terms over a common denominator:
\[
= \left( \frac{2 \sin^2A - \cos^2A}{\sin^2A \cos^4A} \right) - \

Explore the solution to this advanced trigonometric identity: 2/cos^2A − 1/cos^4A − 2/sin^2A + 1/sin^4A = cot^4A - tan^4A. This step-by-step breakdown covers key trigonometric concepts like cotangent, tangent, and the difference of squares identity. Suitable for high school students in Grades 10-12.

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