Let's walk through the problem step by step to understand it clearly.

Problem Recap:

• The triangle $ABC$ is a right-angled triangle at $A$.
• $AB = 6 \, \text{cm}$, $BC = 10 \, \text{cm}$.
• We need to prove that:
  $\sin x \cos y + \cos x \sin y = 1$

Now, let's analyze it in detail.

Step 1: Identify the angles involved.

• $\angle A = 90^\circ$.
• Since $\angle B + \angle C = 90^\circ$ (complementary in a right triangle), we have: $\text{Let } \angle B = x \text{ and } \angle C = y.$ So, $x + y = 90^\circ$.

Step 2: Trigonometric identity to prove.

The given expression is: $\sin x \cos y + \cos x \sin y.$

This is a known trigonometric identity: $\sin(x + y) = \sin x \cos y + \cos x \sin y.$

Since $x + y = 90^\circ$, we have: $\sin(90^\circ) = 1.$ Thus: $\sin x \cos y + \cos x \sin y = 1.$

This proves the given statement.

Step 3: Additional triangle properties.

Let’s ensure all sides are correct using the Pythagorean theorem.
$BC^2 = AB^2 + AC^2.$ With $BC = 10 \, \text{cm}$ and $AB = 6 \, \text{cm}$, we solve for $AC$: $10^2 = 6^2 + AC^2 \implies 100 = 36 + AC^2 \implies AC^2 = 64 \implies AC = 8 \, \text{cm}.$

Summary

1. Since $x + y = 90^\circ$, the expression $\sin x \cos y + \cos x \sin y = \sin(90^\circ) = 1$.
2. This confirms the identity and proves the result.

Do you have any further questions, or would you like more details on a specific part?

Related Questions:

1. What are some other trigonometric identities involving complementary angles?
2. How do you calculate all trigonometric ratios for this triangle?
3. What is the importance of the Pythagorean theorem in right-angled triangles?
4. What are the values of $\sin x$ and $\cos y$ for the given triangle?
5. How does this identity apply to angles beyond $90^\circ$?

Tip: Familiarize yourself with basic trigonometric identities like $\sin(A \pm B)$ to easily solve such problems.