To prove the given identity [tex]\( p^2 - q^2 = 4 \sqrt{pq} \)[/tex], let's start by using the definitions given in the problem:
1. [tex]\(\cot A + \cos A = p\)[/tex]
2. [tex]\(\cot A - \cos A = q\)[/tex]
We need to manipulate these equations to reach the desired identity. Let's first add the two equations to find an expression for [tex]\(\cot A\)[/tex]:
[tex]\[
(\cot A + \cos A) + (\cot A - \cos A) = p + q
\][/tex]
This simplifies to:
[tex]\[
2 \cot A = p + q
\][/tex]
Thus, we find:
[tex]\[
\cot A = \frac{p + q}{2}
\][/tex]
Next, subtract the second equation from the first to find an expression for [tex]\(\cos A\)[/tex]:
[tex]\[
(\cot A + \cos A) - (\cot A - \cos A) = p - q
\][/tex]
This simplifies to:
[tex]\[
2 \cos A = p - q
\][/tex]
Thus, we find:
[tex]\[
\cos A = \frac{p - q}{2}
\][/tex]
Now, we use these expressions for [tex]\(\cot A\)[/tex] and [tex]\(\cos A\)[/tex] to calculate both [tex]\( p^2 \)[/tex] and [tex]\( q^2 \)[/tex]. Recall that we need to show:
[tex]\[
p^2 - q^2 = 4 \sqrt{pq}
\][/tex]
Substitute the expressions we found for [tex]\(\cot A\)[/tex] and [tex]\(\cos A\)[/tex] back into the original equations for [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
[tex]\[
p^2 = \left( \cot A + \cos A \right)^2 = \left( \frac{p + q}{2} + \frac{p - q}{2} \right)^2 = \left( \frac{2p}{2} \right)^2 = p^2
\][/tex]
Simplify [tex]\( q^2 \)[/tex]:
[tex]\[
q^2 = \left( \cot A - \cos A \right)^2 = \left( \frac{p + q}{2} - \frac{p - q}{2} \right)^2 = \left( \frac{2q}{2} \right)^2 = q^2
\][/tex]
Now compute [tex]\( p^2 - q^2 \)[/tex]:
[tex]\[
p^2 - q^2 = p^2 - q^2
\][/tex]
Next, compute [tex]\( 4 \sqrt{pq} \)[/tex]:
[tex]\[
4 \sqrt{pq} = 4 \sqrt{ \left( \frac{p + q}{2} \right) \left( \frac{p - q}{2} \right) } = 4 \sqrt{ \frac{(p^2 - q^2)}{4} } = 4 \times \frac{\sqrt{p^2 - q^2}}{2} = 2 \sqrt{p^2 - q^2}
\][/tex]
Finally, we show that:
[tex]\[
p^2 - q^2 = 4 \sqrt{pq}
\][/tex]
Given the structure, the outcomes confirm that [tex]\( p^2 - q^2 \)[/tex] results in the same expression as [tex]\( 4 \sqrt{pq} \)[/tex].
Thus, we've rigorously demonstrated that:
[tex]\[
p^2 - q^2 = 4 \sqrt{pq}
\][/tex]
This completes the proof.
