Simplify: $\left(\frac{1}{5+\sqrt{3}}\right)+\left(\frac{1}{5-\sqrt{3}}\right)-\left(\frac{10}{25-3}\right)$

The correct option is 0.

Let us simplify the given algebraic expression step-by-step.

The given expression is:

$\left(\frac{1}{5+\sqrt{3}}\right)+\left(\frac{1}{5-\sqrt{3}}\right)-\left(\frac{10}{25-3}\right)$

Step 1: Combine the first two fractions by finding a common denominator.

The common denominator for the first two terms is $\left(5+\sqrt{3}\right)\left(5-\sqrt{3}\right)$.

Adding the two fractions gives:

$\frac{1}{5+\sqrt{3}}+\frac{1}{5-\sqrt{3}}=\frac{\left(5-\sqrt{3}\right)+\left(5+\sqrt{3}\right)}{\left(5+\sqrt{3}\right)\left(5-\sqrt{3}\right)}$

Step 2: Simplify the numerator and the denominator of the combined fraction.

In the numerator, the terms $-\sqrt{3}$ and $+\sqrt{3}$ cancel each other out:

$5-\sqrt{3}+5+\sqrt{3}=10$

In the denominator, apply the algebraic identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$:

$\left(5+\sqrt{3}\right)\left(5-\sqrt{3}\right)=5^2-{\left(\sqrt{3}\right)}^2=25-3$

So, the sum of the first two terms simplifies to:

$\frac{10}{25-3}$

Step 3: Substitute back into the original expression.

Now, subtract the third term from the result:

$\frac{10}{25-3}-\frac{10}{25-3}=0$

Thus, the simplified value of the expression is 0.