The expression \((\sin 45^\circ)^2 + (\cos 45^\circ)^2\) is a standard trigonometric identity. Here's how we solve it:

1. \(\sin 45^\circ = \frac{\sqrt{2}}{2}\)
2. \(\cos 45^\circ = \frac{\sqrt{2}}{2}\)

Now, substitute these values into the expression:

\[
\left( \frac{\sqrt{2}}{2} \right)^2 + \left( \frac{\sqrt{2}}{2} \right)^2
\]

This simplifies to:

\[
\frac{2}{4} + \frac{2}{4} = \frac{4}{4} = 1
\]

Thus, \((\sin 45^\circ)^2 + (\cos 45^\circ)^2 = 1\).

This is consistent with the Pythagorean identity in trigonometry:

\[
\sin^2 \theta + \cos^2 \theta = 1 \quad \text{for any angle} \ \theta.
\]

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. What is the Pythagorean identity in trigonometry?
2. How do sine and cosine relate to right triangles?
3. What are the values of sine and cosine for other common angles (30°, 60°, etc.)?
4. Can the Pythagorean identity be used for angles other than 45°?
5. How can we derive the values of \(\sin 45^\circ\) and \(\cos 45^\circ\) geometrically?

**Tip:** In trigonometry, the identity \(\sin^2 \theta + \cos^2 \theta = 1\) holds true for any angle \(\theta\), not just for 45°.

Discover the solution to the trigonometric identity (sin 45°)² + (cos 45°)² using the Pythagorean Identity. Learn how to simplify the expression using fundamental trigonometric values. Ideal for students in Grades 9-10.

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