---
title: "Euler's theorem for cos⁻¹((x+y)/(√x+√y))"
subject: "Mathematics"
chapter: 1
key: "semester_2/mathematics:euler-s-theorem-cos-inverse"
source: https://www.bctengineer.com/q/semester_2/mathematics/euler-s-theorem-cos-inverse
site: BCT Engineering
---

# Euler's theorem for cos⁻¹((x+y)/(√x+√y))

Mathematics · Chapter 1: Calculus of Two or more variables

- Asked in: 2071 Bhadra, 5 marks
- Video: [395 | If u = cos⁻¹{(x+y)/(√x+√y)} Prove xUx + yUy + (1/2)cot u = 0 | Euler's Theorem | Inverse Trig.](https://www.youtube.com/watch?v=eNy4HiZBu0I) — GAURAV Kumawat (Gk)
- 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-cos-inverse
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-1

## Question

If u = cos⁻¹((x+y)/(√x+√y)) then show that x ∂u/∂x + y ∂u/∂y = −(1/2) cot u.

## Short answer

Let z = cos u = (x+y)/(√x+√y). z is homogeneous of degree 1/2. By Euler’s theorem, x z_x + y z_y = (1/2)z. With z=cos u, this gives −sin u (x u_x + y u_y) = (1/2)cos u, so x u_x + y u_y = −(1/2) cot u.

## Answer

#### Euler's theorem
If f(x, y) is homogeneous of degree *n*, then x ∂f/∂x + y ∂f/∂y = n f.

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

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

Replace x, y by tx, ty: 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

With z = cos u: ∂z/∂x = −sin u ∂u/∂x and ∂z/∂y = −sin u ∂u/∂y. So

```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. For u = cos⁻¹((x+y)/(√x+√y)), Euler's theorem is applied to:
   A. z = u
   B. z = (x+y)/(√x+√y) = cos u
   C. z = sin u
   D. z = cos⁻¹ u
   Answer: B. z = cos u is homogeneous; u is not. (Key point: Function)

2. The degree of z is:
   A. −1/2
   B. 1
   C. 2
   D. 1/2
   Answer: D. z(tx, ty) = t/t^(1/2) z = t^(1/2) z. (Key point: Degree)

3. From −sin u(x u_x + y u_y) = (1/2) cos u, we get:
   A. (1/2) cot u
   B. −(1/2) tan u
   C. −(1/2) cot u
   D. −2 cot u
   Answer: C. Dividing: x u_x + y u_y = −(1/2) cos u/sin u = −(1/2) cot u. (Key point: Result)

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