We have the expression sine squared of x over 2 plus cosine squared of x over 2. This expression follows the fundamental trigonometric identity: sine squared theta plus cosine squared theta equals 1, where theta equals x over 2. Let's visualize this identity using the unit circle.
Now let's apply the fundamental trigonometric identity step by step. First, we recognize that our expression follows the pattern sine squared theta plus cosine squared theta. Second, we apply the identity that this sum always equals 1, where theta equals x over 2. We can visualize this using a right triangle where the hypotenuse has length 1, demonstrating the Pythagorean theorem relationship.
The final answer is 1. This expression equals 1 regardless of the value of x, demonstrating the universal nature of the fundamental trigonometric identity. As we can see in the graph, while sine squared and cosine squared individually vary between 0 and 1, their sum always remains constant at 1. This is a beautiful example of how trigonometric identities provide consistent relationships in mathematics.
让我们用具体数值验证这个恒等式。当x等于π时，我们得到sin²(π/2) + cos²(π/2)，等于1²加0²，结果是1。当x等于π/2时，我们得到sin²(π/4) + cos²(π/4)，等于二分之根号二的平方加二分之根号二的平方，也就是二分之一加二分之一，结果仍是1。单位圆可视化确认了无论角度如何，这个和总是等于1。
In summary, we have shown that sine squared of x over 2 plus cosine squared of x over 2 equals 1. This is a direct application of the fundamental trigonometric identity, which states that sine squared theta plus cosine squared theta equals 1 for any angle theta. The key insight is that this identity represents the Pythagorean theorem applied to the unit circle, where the radius always has length 1. The result is independent of the value of x, demonstrating the universal nature of trigonometric relationships. Therefore, the answer is 1.