How do I check whether these expressions are identical?

Question

Consider:

R0n = (β Λ (-1 + ξ) (-1 +  ρ))/((α + γ + θ + μ) (μ + ξ +  ρ));
xn = σ1^2 + m;
yn = 2 (μ + ρ + ξ) - σ1^2 - m;
k1 = 1/(μ + α + θ + γ);
k2 = Sqrt[R0n]/η;

Then I wish to check whether the following are identical:

(k1 β (1 - ρ) (1 - ξ)/ k2) (xn Λ^2/((μ + ρ + ξ)^2 yn))^(1/2)

and

η ((R0n xn)/yn)^(1/2)

I tried FullSimplify but it is still a mess and didn't make things clearer.

Edit:

Applying FullSimplify to the first expression, we get:

 (β η (-1 + ξ) (-1 + ρ) Sqrt[-((Λ^2 \
(m + σ1^2))/((μ + ξ + ρ)^2 (m - 
     2 (μ + ξ + ρ) + σ1^2)))])/Sqrt[(β \
Λ (α + γ + θ + μ) (-1 + \
ξ) (-1 + ρ))/(μ + ξ + ρ)]

Applying it to the second:

η Sqrt[-((β Λ (-1 + ξ) (-1 + ρ) (m \
+ σ1^2))/((α + γ + θ + μ) (μ + \
ξ + ρ) (m - 2 (μ + ξ + ρ) + σ1^2)))]

Now shifting β (-1 + ξ) (-1 + ρ) into the radical, and applying Fullsimplify to inside of the radical, we get:

-((β Λ (-1 + ξ) (-1 + ρ) (m + \
σ1^2))/((α + γ + θ + μ) (μ + \
ξ + ρ) (m - 2 (μ + ξ + ρ) + σ1^2)))

which is identical to the inside of the radical of the second expression.

Answer 1

$Version

(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)

Clear["Global`*"]

R0n = (β Λ (-1 + ξ) (-1 + ρ))/((α + \
γ + θ + μ) (μ + ξ + ρ));
xn = σ1^2 + m;
yn = 2 (μ + ρ + ξ) - σ1^2 - m;
k1 = 1/(μ + α + θ + γ);
k2 = Sqrt[R0n]/η;

var = {m, α, β, γ, η, θ, Λ, μ, ξ, ρ, σ1};

You need to specify some assumptions or constraints on the variables. For example,

Assuming[Element[var, PositiveReals] && ρ > 1 && ξ > 1,
 (k1 β (1 - ρ) (1 - ξ)/
       k2) (xn Λ^2/((μ + ρ + ξ)^2 yn))^(1/
       2) == η ((R0n xn)/yn)^(1/2) // FullSimplify]

(* True *)

EDIT: Or, another example

Assuming[
 Element[var, PositiveReals] && 0 < ξ < 1 && 0 < ρ < 1, 
    (k1 β (1 - ρ) (1 - ξ)/
       k2) (xn Λ^2/((μ + ρ + ξ)^2 yn))^(1/
       2) == η ((R0n xn)/yn)^(1/2) // FullSimplify]

(* True *)

Answer 2

Here is a Monte Carlo way to verify this.

You may choose random positive reals for the parameters and check if the outcome is true. You then repeat this many times. And if you always get True the probability of the claim to be true goes towards 1:

AllTrue[Table[
  e1 == e2 /. 
   Thread[{\[Mu], \[Xi], \[Rho] , m, \[Sigma]1, k1, k2, xn, 
      yn, \[CapitalLambda]} ->  RandomReal[{0, 100}, 10]], 100], # &]

(* True *)