Let's solve the given equation step by step:
[tex]\[
\frac{(1+\cos A)^2 - (1-\cos A)^2}{\sin^2 A} = 4 \operatorname{cosec} A \cdot \cot A
\][/tex]
First, let's simplify the expression on the left-hand side.
### Step 1: Expand the Numerator
We start with the numerator [tex]\((1 + \cos A)^2 - (1 - \cos A)^2\)[/tex].
Using the algebraic identity [tex]\((a + b)^2 - (a - b)^2 = 4ab\)[/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 these two expanded forms:
[tex]\[
(1 + \cos A)^2 - (1 - \cos A)^2 = (1 + 2 \cos A + \cos^2 A) - (1 - 2 \cos A + \cos^2 A)
\][/tex]
[tex]\[
= 1 + 2 \cos A + \cos^2 A - 1 + 2 \cos A - \cos^2 A
\][/tex]
[tex]\[
= (1 - 1) + (2 \cos A + 2 \cos A) + (\cos^2 A - \cos^2 A)
\][/tex]
[tex]\[
= 4 \cos A
\][/tex]
### Step 2: Form the Fraction
Now, place the result back in the original fraction:
[tex]\[
\frac{4 \cos A}{\sin^2 A}
\][/tex]
### Step 3: Work with Trigonometric Identities on the Right-hand Side
The given right-hand side is [tex]\(4 \operatorname{cosec} A \cdot \cot A\)[/tex].
Recall the trigonometric identities:
[tex]\[
\operatorname{cosec} A = \frac{1}{\sin A}
\][/tex]
[tex]\[
\cot A = \frac{\cos A}{\sin A}
\][/tex]
So,
[tex]\[
4 \operatorname{cosec} A \cdot \cot A = 4 \left(\frac{1}{\sin A}\right) \left(\frac{\cos A}{\sin A}\right)
\][/tex]
### Step 4: Simplify the Right-hand Side
Multiply these fractions:
[tex]\[
4 \cdot \frac{1}{\sin A} \cdot \frac{\cos A}{\sin A} = 4 \cdot \frac{\cos A}{\sin^2 A}
\][/tex]
### Step 5: Comparison
The left-hand side simplifies to:
[tex]\[
\frac{4 \cos A}{\sin^2 A}
\][/tex]
The right-hand side simplifies to:
[tex]\[
\frac{4 \cos A}{\sin^2 A}
\][/tex]
Both sides are equal. Thus, we can conclude:
[tex]\[
\frac{(1+\cos A)^2 - (1-\cos A)^2}{\sin^2 A} = 4 \operatorname{cosec} A \cdot \cot A
\][/tex]
And thereby, the trigonometric identity holds true.
