The mathematical expression in the image is:

$\prod_{n=1}^{k-1} \sin\left(\frac{(2n+1)\pi}{2}\right)$

Let's evaluate this expression step by step using concepts from trigonometry and number theory.

Step 1: Analyze the sine term

The term inside the sine function is:

$\frac{(2n+1)\pi}{2}$

For various $n$, this expression evaluates to odd multiples of $\frac{\pi}{2}$. Recall the values of $\sin(x)$ for odd multiples of $\frac{\pi}{2}$:

• $\sin\left(\frac{\pi}{2}\right) = 1$
• $\sin\left(\frac{3\pi}{2}\right) = -1$
• $\sin\left(\frac{5\pi}{2}\right) = 1$
• $\sin\left(\frac{7\pi}{2}\right) = -1$, and so on.

The sine function alternates between $1$ and $-1$ for consecutive odd multiples of $\frac{\pi}{2}$.

Step 2: Behavior of the product

Since the sine values alternate between $1$ and $-1$, the product alternates in sign. Specifically, for $k-1$ terms in the product:

• If $k-1$ is even, there are an equal number of $1$s and $-1$s, resulting in: $\prod_{n=1}^{k-1} \sin\left(\frac{(2n+1)\pi}{2}\right) = 1$
• If $k-1$ is odd, there will be one extra $-1$, leading to: $\prod_{n=1}^{k-1} \sin\left(\frac{(2n+1)\pi}{2}\right) = -1$

Step 3: Final result

The value of the product depends on whether $k-1$ is even or odd:

• If $k-1$ is even, the result is $1$.
• If $k-1$ is odd, the result is $-1$.

Final Answer: