Find the exact value of the trigonometric expression: cos^2 20° + cos^2 70° Choose the correct exact value.

Introduction / Context: This question tests complementary-angle identities and the Pythagorean identity. Angles 20° and 70° are complementary (they add to 90°), which allows rewriting cosine in terms of sine and simplifying easily.

Given Data / Assumptions:

• Expression: cos^2 20° + cos^2 70° • Use standard identities for complementary angles

Concept / Approach: Use the complementary identity: cos(90° − θ) = sin θ. Since 70° = 90° − 20°, we have: cos 70° = sin 20°. Then: cos^2 70° = sin^2 20°. So the expression becomes: cos^2 20° + sin^2 20°, which equals 1 by the Pythagorean identity.

Step-by-Step Solution: 1) Note that 70° = 90° − 20° 2) Apply identity: cos(90° − θ) = sin θ cos 70° = sin 20° 3) Square both sides: cos^2 70° = sin^2 20° 4) Substitute into the expression: cos^2 20° + cos^2 70° = cos^2 20° + sin^2 20° 5) Use identity: sin^2 θ + cos^2 θ = 1 6) Therefore the value is 1.

Verification / Alternative check: A quick numerical sense-check: cos 20° ≈ 0.94 so cos^2 20° ≈ 0.88. cos 70° ≈ 0.34 so cos^2 70° ≈ 0.12. Sum ≈ 1.00, consistent with the exact result 1.

Why Other Options Are Wrong: • 0 or 2: impossible because each squared cosine is between 0 and 1, and the pair are complementary giving a Pythagorean sum. • 1/2 or 13: not consistent with the identity-based simplification.

Common Pitfalls: • Forgetting that 70° is complementary to 20°. • Using cos(90° − θ) = cos θ (incorrect).

Final Answer: 1