Certainly! Let's solve the given expression step-by-step.
We need to verify the equation:
[tex]\[
\frac{(1 + \cos A)^2 - (1 - \cos A)^2}{\sin^2 A} = 4 \csc A \cdot \cot A
\][/tex]
### Step 1: Expand the Numerator
First, let’s simplify the numerator [tex]\((1 + \cos A)^2 - (1 - \cos A)^2\)[/tex].
[tex]\[
(1 + \cos A)^2 = 1 + 2\cos A + \cos^2 A
\][/tex]
[tex]\[
(1 - \cos A)^2 = 1 - 2\cos A + \cos^2 A
\][/tex]
Subtracting the two expressions, we get:
[tex]\[
(1 + \cos A)^2 - (1 - \cos A)^2 = (1 + 2\cos A + \cos^2 A) - (1 - 2\cos A + \cos^2 A)
\][/tex]
Combine like terms:
[tex]\[
= 1 + 2\cos A + \cos^2 A - 1 + 2\cos A - \cos^2 A
\][/tex]
[tex]\[
= 4\cos A
\][/tex]
### Step 2: Formulate the Left-Hand Side (LHS)
Now, our expression on the left-hand side becomes:
[tex]\[
\frac{4 \cos A}{\sin^2 A}
\][/tex]
So, the left-hand side (LHS) is:
[tex]\[
\frac{4 \cos A}{\sin^2 A}
\][/tex]
### Step 3: Simplify for the Right-Hand Side (RHS)
We need to verify that this equals [tex]\(4 \csc A \cdot \cot A\)[/tex]. Let’s rewrite the right-hand side using the definitions of cosecant and cotangent:
[tex]\[
\csc A = \frac{1}{\sin A}, \quad \cot A = \frac{\cos A}{\sin A}
\][/tex]
Thus,
[tex]\[
4 \csc A \cdot \cot A = 4 \left(\frac{1}{\sin A}\right) \left(\frac{\cos A}{\sin A}\right)
\][/tex]
[tex]\[
= 4 \cdot \frac{\cos A}{\sin^2 A}
\][/tex]
### Step 4: Equate LHS and RHS
We see that both sides match:
[tex]\[
\frac{4 \cos A}{\sin^2 A} = 4 \csc A \cdot \cot A
\][/tex]
Therefore, the given expression is indeed an identity:
[tex]\[
\boxed{\frac{(1 + \cos A)^2 - (1 - \cos A)^2}{\sin^2 A} = 4 \csc A \cdot \cot A}
\][/tex]
