Let's solve the given equation step by step and verify the equality:
[tex]\[ \frac{1 - \cos^4 A}{\sin^4 A} = 1 + 2 \cot^2 A \][/tex]
### Step 1: Simplify the left-hand side
First, simplify the left-hand side of the equation. Rewrite it in terms of [tex]\(\sin A\)[/tex] and [tex]\(\cos A\)[/tex]:
[tex]\[ \frac{1 - \cos^4 A}{\sin^4 A} \][/tex]
We know that [tex]\(\sin^2 A + \cos^2 A = 1\)[/tex]. Let’s use this identity to help simplify the expression. Rewrite [tex]\(\cos^4 A\)[/tex] as [tex]\((\cos^2 A)^2\)[/tex]:
[tex]\[ \frac{1 - (\cos^2 A)^2}{\sin^4 A} \][/tex]
Using the difference of squares, we can factor the numerator:
[tex]\[ 1 - (\cos^2 A)^2 = (1 - \cos^2 A)(1 + \cos^2 A) \][/tex]
Since [tex]\(1 - \cos^2 A = \sin^2 A\)[/tex], the expression becomes:
[tex]\[ \frac{\sin^2 A (1 + \cos^2 A)}{\sin^4 A} \][/tex]
Simplify by cancelling [tex]\(\sin^2 A\)[/tex] in the numerator with [tex]\(\sin^4 A\)[/tex] in the denominator:
[tex]\[ \frac{1 + \cos^2 A}{\sin^2 A} \][/tex]
### Step 2: Compare with the right-hand side
Next, let’s rewrite the right-hand side using the definition of [tex]\(\cot A\)[/tex]:
[tex]\[ 1 + 2 \cot^2 A \][/tex]
We know that [tex]\(\cot A = \frac{\cos A}{\sin A}\)[/tex], so:
[tex]\[ \cot^2 A = \left(\frac{\cos A}{\sin A}\right)^2 = \frac{\cos^2 A}{\sin^2 A} \][/tex]
Substituting this into the equation:
[tex]\[ 1 + 2 \cot^2 A = 1 + 2 \frac{\cos^2 A}{\sin^2 A} \][/tex]
Combine the fraction:
[tex]\[ 1 + 2 \left(\frac{\cos^2 A}{\sin^2 A}\right) = \frac{\sin^2 A}{\sin^2 A} + \frac{2 \cos^2 A}{\sin^2 A} \][/tex]
Combine terms under a common denominator:
[tex]\[ \frac{\sin^2 A + 2 \cos^2 A}{\sin^2 A} \][/tex]
### Step 3: Verifying the equality
We need to check if these expressions are equal:
[tex]\[ \frac{1 + \cos^2 A}{\sin^2 A} \overset{?}{=} \frac{\sin^2 A + 2 \cos^2 A}{\sin^2 A} \][/tex]
Note that:
[tex]\[ 1 = \sin^2 A + \cos^2 A \][/tex]
Use this to simplify the expression on the right-hand side:
[tex]\[ \frac{\sin^2 A + 2 \cos^2 A}{\sin^2 A} = \frac{(\sin^2 A + \cos^2 A) + \cos^2 A}{\sin^2 A} = \frac{1 + \cos^2 A}{\sin^2 A} \][/tex]
Thus, the simplified left-hand side and right-hand side expressions are equal:
[tex]\[ \frac{1 + \cos^2 A}{\sin^2 A} = \frac{1 + \cos^2 A}{\sin^2 A} \][/tex]
Therefore, the original equation:
[tex]\[ \frac{1-\cos^4 A}{\sin^4 A} = 1 + 2 \cot^2 A \][/tex]
is indeed correct.
