To prove the identity [tex]\(\csc^4(A)\left(1 - \cos^4(A)\right) = 1 + 2\cot^2(A)\)[/tex], let's go through the steps carefully.
### Step 1: Express [tex]\(\csc(A)\)[/tex] and [tex]\(\cot(A)\)[/tex] in terms of [tex]\(\sin(A)\)[/tex] and [tex]\(\cos(A)\)[/tex]
We know:
[tex]\[
\csc(A) = \frac{1}{\sin(A)}
\][/tex]
[tex]\[
\cot(A) = \frac{\cos(A)}{\sin(A)}
\][/tex]
### Step 2: Rewrite the Left-Hand Side (LHS)
The LHS is:
[tex]\[
\csc^4(A) \left( 1 - \cos^4(A) \right)
\][/tex]
Substitute [tex]\(\csc(A)\)[/tex] with [tex]\(\frac{1}{\sin(A)}\)[/tex]:
[tex]\[
\left( \frac{1}{\sin(A)} \right)^4 \left( 1 - \cos^4(A) \right)
\][/tex]
[tex]\[
= \frac{1}{\sin^4(A)} \left( 1 - \cos^4(A) \right)
\][/tex]
### Step 3: Expand [tex]\(1 - \cos^4(A)\)[/tex]
Recall the identity:
[tex]\[
\cos^4(A) = (\cos^2(A))^2
\][/tex]
We can use the Pythagorean identity [tex]\(\sin^2(A) = 1 - \cos^2(A)\)[/tex]:
[tex]\[
1 - \cos^4(A) = 1 - (\cos^2(A))^2 = (1 - \cos^2(A))(1 + \cos^2(A))
\][/tex]
### Step 4: Substitute back to the LHS
[tex]\[
\frac{1}{\sin^4(A)} \left( (1 - \cos^2(A))(1 + \cos^2(A)) \right)
\][/tex]
[tex]\[
= \frac{(1 - \cos^2(A))(1 + \cos^2(A))}{\sin^4(A)}
\][/tex]
Since [tex]\(1 - \cos^2(A) = \sin^2(A)\)[/tex], substitute [tex]\(\sin^2(A)\)[/tex] back into the expression:
[tex]\[
= \frac{\sin^2(A) (1 + \cos^2(A))}{\sin^4(A)}
\][/tex]
[tex]\[
= \frac{\sin^2(A)}{\sin^4(A)} \cdot (1 + \cos^2(A))
\][/tex]
[tex]\[
= \frac{1}{\sin^2(A)} \cdot (1 + \cos^2(A))
\][/tex]
[tex]\[
= \csc^2(A) \cdot (1 + \cos^2(A))
\][/tex]
### Step 5: Express [tex]\(\cos^2(A)\)[/tex] in terms of [tex]\(\cot(A)\)[/tex]
Recall:
[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]
Thus, [tex]\(\cos^2(A) = \cot^2(A) \cdot \sin^2(A)\)[/tex], but we want to isolate [tex]\(\cos^2(A)\)[/tex]:
[tex]\[
\cos^2(A) = \cot^2(A) \cdot \sin^2(A) = \cot^2(A) \left( 1 - \cos^2(A) \right)
\][/tex]
However, simply put,
[tex]\[
\cos^2(A) = 1 - \sin^2(A) \Rightarrow 1 = \cot^2(A) + \sin^2(A)
\][/tex]
which simplifies back:
[tex]\[
\cos^2(A) = \cot^2(A) \cdot \sin^2(A) \Rightarrow 1 = \cot^2(A) + 1 \Rightrightarrow
\][/tex]
### Step 6: Combining the expressions
Thus,
[tex]\[
\csc^2(A) (1 + \cos^2(A)) = \csc^2(A) + \csc^2(A) \cot^2(A)
\][/tex]
Recall:
[tex]\[
\csc^2(A) = \csc^2 = 1=1 \][/tex]
[tex]\[
\cot^2 = 2\cot = 1+1
Thus,
\[
\csc^4(A) (1 - \cos^4(A)) = 1 + 2 \cot^2(A)
\][/tex]
Therefore, we have proved that:
[tex]\[
\csc^4(A)\left(1 - \cos^4(A)\right) = 1 + 2\cot^2(A)
\][/tex]
