$\text{If}\tan \left(t\right)+\cot \left(t\right)=1\text{, then one of the values of the expression}\frac{1}{\sin \left(t\right)+\cos \left(t\right)}\text{is}\text{\_\_\_\_\_}.$

The correct answer is $\frac{1}{\sqrt{3}}$.

Step 1: Simplify the given equation

We are given the trigonometric equation:
$\tan \left(t\right)+\cot \left(t\right)=1$
Expressing tangent and cotangent in terms of sine and cosine:
$\frac{\sin \left(t\right)}{\cos \left(t\right)}+\frac{\cos \left(t\right)}{\sin \left(t\right)}=1$
Combining the fractions using a common denominator:
$\frac{{\sin }^2\left(t\right)+{\cos }^2\left(t\right)}{\sin \left(t\right)\cos \left(t\right)}=1$
Applying the fundamental Pythagorean trigonometric identity:
${\sin }^2\left(t\right)+{\cos }^2\left(t\right)=1$
This simplifies the equation to:
$\frac{1}{\sin \left(t\right)\cos \left(t\right)}=1$
Cross-multiplying gives:
$\sin \left(t\right)\cos \left(t\right)=1$

Step 2: Find the value of sum of sine and cosine

Consider the square of the expression:
$\left(\sin \right(t)+\cos (t){)}^2$
Expanding using the algebraic identity:
$(a+b{)}^2=a^2+b^2+2ab$
We get:
$\left(\sin \right(t)+\cos (t){)}^2={\sin }^2(t)+{\cos }^2(t)+2\sin (t\left)\cos \right(t)$
Substitute the known values:
${\sin }^2\left(t\right)+{\cos }^2\left(t\right)=1$
and:
$\sin \left(t\right)\cos \left(t\right)=1$
So:
$\left(\sin \right(t)+\cos (t){)}^2=1+2(1)=3$
Taking the square root on both sides:
$\sin \left(t\right)+\cos \left(t\right)=\pm \sqrt{3}$

Step 3: Calculate the required expression

We are asked to evaluate:
$\frac{1}{\sin \left(t\right)+\cos \left(t\right)}$
Substituting the value obtained for the denominator:
$\frac{1}{\sin \left(t\right)+\cos \left(t\right)}=\frac{1}{\pm \sqrt{3}}=\pm \frac{1}{\sqrt{3}}$
Therefore, one of the possible values is:
$\frac{1}{\sqrt{3}}$