To calculate $\cos 25^\circ$ and $\cos 73^\circ$ using Newton's interpolation formula, we'll proceed with the following steps:

Newton’s Forward Difference Interpolation Formula:

The Newton forward difference interpolation formula is given as:

$f(x) = f(x_0) + \frac{(x - x_0)}{1!} \Delta f(x_0) + \frac{(x - x_0)(x - x_1)}{2!} \Delta^2 f(x_0) + \cdots$

Where:

• $f(x)$ is the interpolated value of the function.
• $x_0$ is the first value of $x$ in the table.
• $\Delta f(x_0)$, $\Delta^2 f(x_0)$, etc., are the forward differences.

We'll calculate both $\cos 25^\circ$ and $\cos 73^\circ$ by applying this method using the data provided in the table: