If $\cot \theta =\frac{28}{45}$, then find the value of $(\sin \theta -\cos \theta )$.

The correct option is 17/53.

Given:

$\cot \theta =\frac{28}{45}$

In a right-angled triangle, the cotangent ratio is defined as the ratio of the length of the adjacent side to the opposite side:

$\cot \theta =\frac{\text{Adjacent}}{\text{Opposite}}=\frac{28}{45}$

Therefore, we can consider:

Adjacent side = 28

Opposite side = 45

Using the Pythagorean theorem, we calculate the hypotenuse:

$\text{Hypotenuse}=\sqrt{{\text{Opposite}}^2+{\text{Adjacent}}^2}$

$\text{Hypotenuse}=\sqrt{{45}^2+{28}^2}$

$\text{Hypotenuse}=\sqrt{2025+784}$

$\text{Hypotenuse}=\sqrt{2809}=53$

Next, we determine the values of $\sin \theta$ and $\cos \theta$:

$\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}=\frac{45}{53}$

$\cos \theta =\frac{\text{Adjacent}}{\text{Hypotenuse}}=\frac{28}{53}$

Now, calculate $(\sin \theta -\cos \theta )$:

$\sin \theta -\cos \theta =\frac{45}{53}-\frac{28}{53}$

$\sin \theta -\cos \theta =\frac{45-28}{53}=\frac{17}{53}$

Hence, the value of $(\sin \theta -\cos \theta )$ is 17/53.