Sure! Let's start with the given equations:
[tex]\[
x = \tan \theta + \sin \theta
\][/tex]
[tex]\[
y = \tan \theta - \sin \theta
\][/tex]
We need to show that [tex]\(x^2 - y^2 = 4 \sqrt{xy}\)[/tex].
1. Square both equations:
[tex]\[
x^2 = (\tan \theta + \sin \theta)^2
\][/tex]
[tex]\[
y^2 = (\tan \theta - \sin \theta)^2
\][/tex]
2. Expand the squares:
[tex]\[
x^2 = \tan^2 \theta + 2 \tan \theta \sin \theta + \sin^2 \theta
\][/tex]
[tex]\[
y^2 = \tan^2 \theta - 2 \tan \theta \sin \theta + \sin^2 \theta
\][/tex]
3. Subtract the second equation from the first:
[tex]\[
x^2 - y^2 = (\tan^2 \theta + 2 \tan \theta \sin \theta + \sin^2 \theta) - (\tan^2 \theta - 2 \tan \theta \sin \theta + \sin^2 \theta)
\][/tex]
4. Simplify the expression:
[tex]\[
x^2 - y^2 = ( \tan^2 \theta + \sin^2 \theta ) + 2 \tan \theta \sin \theta - ( \tan^2 \theta + \sin^2 \theta ) + 2 \tan \theta \sin \theta
\][/tex]
[tex]\[
x^2 - y^2 = 4 \tan \theta \sin \theta
\][/tex]
5. Find an expression for [tex]\(xy\)[/tex] by multiplying [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
xy = (\tan \theta + \sin \theta)(\tan \theta - \sin \theta)
\][/tex]
6. Expand the product:
[tex]\[
xy = \tan^2 \theta - \sin^2 \theta
\][/tex]
7. Use the identity for [tex]\(\tan^2 \theta\)[/tex]:
Recall that [tex]\( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\)[/tex], so:
[tex]\[
xy = \frac{\sin^2 \theta}{\cos^2 \theta} - \sin^2 \theta = \sin^2 \theta \left(\frac{1}{\cos^2 \theta} - 1 \right)
\][/tex]
[tex]\[
xy = \sin^2 \theta \left(\frac{1 - \cos^2 \theta}{\cos^2 \theta} \right)
\][/tex]
[tex]\[
xy = \sin^2 \theta \left(\frac{\sin^2 \theta}{\cos^2 \theta} \right)
\][/tex]
[tex]\[
xy = \frac{\sin^4 \theta}{\cos^2 \theta}
\][/tex]
8. Simplify:
[tex]\[
\sqrt{xy} = \sqrt{\frac{\sin^4 \theta}{\cos^2 \theta}} = \frac{\sin^2 \theta}{\cos \theta}
\][/tex]
9. Find [tex]\(4 \sqrt{xy}\)[/tex]:
[tex]\[
4 \sqrt{xy} = 4 \cdot \frac{\sin^2 \theta}{\cos \theta} = \frac{4 \tan \theta \sin \theta}{\zeta}
\][/tex]
Combining all pieces, we have shown that
[tex]\[
x^2 - y^2 = 4 \sqrt{xy}
\][/tex]
Hence, the identity is proved.
