Explain this briefly ### Mathematical Proof of a Trigonometric Identity#### Given:\[ \tan u = \frac{\partial u / \partial y}{\partial u / \partial x} \]#### To Prove:\[ \left(\frac{\partial r^{2}}{\partial x}\right)^{2} + \left(\frac{\partial r^{2}}{\partial y}\right)^{2} = r^{2} \left(\frac{\partial r^{2}}{\partial y}\right)\]In this proof, we start with the trigonometric function tangent (tan), which is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. The given equation involves partial derivatives of a function `u` with respect to the variables `x` and `y`.**Explanation of Work:**1. **Original Equation:**\[ \tan u = \frac{\frac{\partial u}{\partial y}}{\frac{\partial u}{\partial x}} \]2. **Objective:**\[ \left( \frac{\partial r^{2}}{\partial x} \right)^{2} + \left( \frac{\partial r^{2}}{\partial y} \right)^{2} = r^{2} \left( \frac{\partial r^{2}}{\partial y} \right) \]*Note: Explanation assumes partial derivatives and uses squared terms to potentially simplify and manipulate to the desired equation.*In summary, by analyzing the given function and using differentiation techniques, we aim to validate the given identity through a series of mathematical manipulations. This involves understanding the relationship between the trigonometric function and partial derivatives in a geometrical context.

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Description: Explain this briefly ### Mathematical Proof of a Trigonometric Identity#### Given:\[ \tan u = \frac{\partial u / \partial y}{\partial u / \partial x} \]#### To Prove:\[ \left(\frac{\partial r^{2}}{\partial x}\right)^{2} + \left(\frac{\partial r^{2}}{

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x+y uis = 1 n upi = no • prove that x-+ y-

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Explain this briefly

### Mathematical Proof of a Trigonometric Identity

#### Given:
\[ \tan u = \frac{\partial u / \partial y}{\partial u / \partial x} \]

#### To Prove:
\[ \left(\frac{\partial r^{2}}{\partial x}\right)^{2} + \left(\frac{\partial r^{2}}{\partial y}\right)^{2} = r^{2} \left(\frac{\partial r^{2}}{\partial y}\right)\]

In this proof, we start with the trigonometric function tangent (tan), which is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. The given equation involves partial derivatives of a function `u` with respect to the variables `x` and `y`.

**Explanation of Work:**

1. **Original Equation:**

\[ \tan u = \frac{\frac{\partial u}{\partial y}}{\frac{\partial u}{\partial x}} \]

2. **Objective:**

\[ \left( \frac{\partial r^{2}}{\partial x} \right)^{2} + \left( \frac{\partial r^{2}}{\partial y} \right)^{2} = r^{2} \left( \frac{\partial r^{2}}{\partial y} \right) \]

*Note: Explanation assumes partial derivatives and uses squared terms to potentially simplify and manipulate to the desired equation.*

In summary, by analyzing the given function and using differentiation techniques, we aim to validate the given identity through a series of mathematical manipulations. This involves understanding the relationship between the trigonometric function and partial derivatives in a geometrical context.

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