Establish the​ identity, left parenthesis sine theta cosine variant phi right parenthesis squared plus left parenthesis sine theta sine variant phi right parenthesis squared plus cosine squared theta equals 1.

We are tasked with proving the identity:

\[
(\sin\theta \cos\varphi)^2 + (\sin\theta \sin\varphi)^2 + \cos^2\theta = 1
\]

### Step-by-step solution:

1. **Expand the squares in the first two terms:**

\[
(\sin\theta \cos\varphi)^2 = \sin^2\theta \cos^2\varphi
\]
\[
(\sin\theta \sin\varphi)^2 = \sin^2\theta \sin^2\varphi
\]

So the original expression becomes:

\[
\sin^2\theta \cos^2\varphi + \sin^2\theta \sin^2\varphi + \cos^2\theta
\]

2. **Factor out \(\sin^2\theta\) from the first two terms:**

\[
\sin^2\theta (\cos^2\varphi + \sin^2\varphi) + \cos^2\theta
\]

3. **Use the Pythagorean identity \(\cos^2\varphi + \sin^2\varphi = 1\):**

\[
\sin^2\theta (1) + \cos^2\theta
\]

This simplifies to:

\[
\sin^2\theta + \cos^2\theta
\]

4. **Use the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\):**

Thus, we have:

\[
1 = 1
\]

### Conclusion:

We have established that the given identity is true.

---
Would you like more details, or do you have any questions? 

Here are 5 related questions to extend your understanding:
1. How can you derive the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\)?
2. Can you prove a similar identity for tangent and secant functions?
3. How does this identity relate to the unit circle in trigonometry?
4. How can you use this identity in solving trigonometric equations?
5. What happens if you replace \(\theta\) and \(\varphi\) with specific angles (like \(\theta = 30^\circ\), \(\varphi = 45^\circ\))?

**Tip:** Memorizing the Pythagorean identities will help simplify many trigonometric problems quickly!

Establish the identity (sine theta cosine variant phi) squared plus (sine theta sine variant phi) squared plus cosine squared theta equals 1.

Learn how to prove the trigonometric identity (sin(θ)cos(ϕ))^2 + (sin(θ)sin(ϕ))^2 + cos^2(θ) = 1 using Pythagorean identities. This problem explores key trigonometric concepts and is suitable for students in Grades 10-12.

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