---
title: "Euler's theorem statement with cos⁻¹ example"
subject: "Mathematics"
chapter: 1
key: "semester_2/mathematics:euler-s-theorem-statement-cos"
source: https://www.bctengineer.com/q/semester_2/mathematics/euler-s-theorem-statement-cos
site: BCT Engineering
---

# Euler's theorem statement with cos⁻¹ example

Mathematics · Chapter 1: Calculus of Two or more variables

- Asked in: 2068 Bhadra, 1+4 marks
- Video: [(English)ENA || Question 9.7 || Euler's Theorem || Proof of Euler's Theorem](https://www.youtube.com/watch?v=riS5T4pnDjI) — Electrical Engineering Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/euler-s-theorem-statement-cos
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-1

## Question

State Euler's theorem for homogeneous function of two variables. If u = cos⁻¹((x+y)/(√x+√y)), then prove that x ∂u/∂x + y ∂u/∂y = −(1/2) cot u.

## Short answer

Euler’s theorem: If f(tx,ty)=tⁿf(x,y) with continuous first partials, then x f_x + y f_y = n f. Here z=cos u=(x+y)/(√x+√y) is homogeneous of degree 1/2, so x z_x+y z_y=z/2. Since z_x=−sin u u_x, similar for y, we get −sin u (x u_x+y u_y)=(1/2)cos u, giving x ∂u/∂x+y ∂u/∂y=−(1/2)cot u.

## Answer

#### Euler's theorem (1 mark)
If f(x, y) is a homogeneous function of degree *n*, i.e. f(tx, ty) = tⁿ f(x, y), with continuous first partial derivatives, then

x ∂f/∂x + y ∂f/∂y = n f.

#### Proof (4 marks)
Given u = cos⁻¹((x + y)/(√x + √y)). Let

z = cos u = (x + y)/(x^(1/2) + y^(1/2)).

Then z(tx, ty) = t(x + y)/(t^(1/2)(√x + √y)) = t^(1/2) z, so z is homogeneous of degree n = 1/2.

By Euler's theorem:

x ∂z/∂x + y ∂z/∂y = (1/2) z

Since z = cos u: ∂z/∂x = −sin u ∂u/∂x, ∂z/∂y = −sin u ∂u/∂y. Therefore

```text
−sin u ( x ∂u/∂x + y ∂u/∂y ) = (1/2) cos u
x ∂u/∂x + y ∂u/∂y = −(1/2) cos u / sin u = −(1/2) cot u
```

Hence proved. ∎

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Euler's theorem for homogeneous f of degree n gives:
   A. x f_x + y f_y = n²f
   B. x f_x + y f_y = n f
   C. f_x + f_y = n
   D. x f_x + y f_y = f
   Answer: B. Standard statement. (Key point: Statement)

2. z = cos u = (x+y)/(√x+√y) has degree:
   A. 1
   B. 2
   C. 1/2
   D. 3/2
   Answer: C. t/t^(1/2) = t^(1/2). (Key point: Degree)

3. The result is:
   A. −cot u
   B. (1/2) cot u
   C. −(1/2) tan u
   D. −(1/2) cot u
   Answer: D. -sin u(x u_x+y u_y) = (1/2)cos u. (Key point: Result)

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