In right angled triangle ABC, angle A = 90°. What is the value of cos²A + cos²B + cos²C?

Introduction / Context:
This conceptual trigonometry question uses basic properties of angles in a triangle and the relationships between trigonometric functions of complementary angles. It asks for the sum cos²A + cos²B + cos²C in a right angled triangle, which is a neat identity that you can derive from simpler facts about right triangles.

Given Data / Assumptions:

• Triangle ABC is right angled at A, so angle A = 90°.
• Therefore, angles B and C are acute and satisfy B + C = 90°.
• We must compute cos²A + cos²B + cos²C.
• Standard trigonometric identities for complementary angles apply: if B + C = 90°, then C = 90° − B.

Concept / Approach:
Use the fact that the cosine of 90° is zero, so cosA becomes very simple. Then, exploit that B and C are complementary. For complementary angles, cos and sin are co functions: cos(90° − θ) = sinθ. This leads to cosC = sinB and cos²C = sin²B. Combining cos²B and sin²B gives 1 due to the fundamental Pythagorean identity for trigonometric functions.

Step-by-Step Solution:
Step 1: Since A = 90°, cosA = cos90° = 0, so cos²A = 0² = 0. Step 2: We know B + C = 90°, hence C = 90° − B. Step 3: Using co function relation, cosC = cos(90° − B) = sinB. Step 4: Therefore cos²C = (sinB)² = sin²B. Step 5: Now consider cos²B + cos²C. Substitute cos²C = sin²B to get cos²B + sin²B. Step 6: By the basic identity sin²B + cos²B = 1 for any angle B, we obtain cos²B + cos²C = 1. Step 7: Finally, add cos²A: cos²A + cos²B + cos²C = 0 + 1 = 1.

Verification / Alternative check:
Pick a specific right triangle, for example with angles A = 90°, B = 30° and C = 60°. Then cos²A = cos²90° = 0, cos²B = cos²30° = (√3 / 2)² = 3 / 4 and cos²C = cos²60° = (1 / 2)² = 1 / 4. Adding gives 0 + 3 / 4 + 1 / 4 = 1. Any other valid right triangle angle pair will also satisfy the same identity because it rests on the general relation sin²θ + cos²θ = 1 and the complementary angle property.

Why Other Options Are Wrong:
2 and 3 are too large and would require cos²B and cos²C to sum to more than 1, which is impossible since each cosine square is at most 1. 0 would imply cos²B and cos²C somehow cancel, which cannot happen with non negative squares. The value −1 is impossible because a sum of squares cannot be negative.

Common Pitfalls:
One common mistake is to confuse cos²A (cosine squared) with cos(2A) (cosine of double angle). Another error is to forget that B and C are complementary and miss the link cosC = sinB. Always check carefully whether the question uses a power (cos²) or a multiple angle (cos 2θ), and use the identity sin²θ + cos²θ = 1 whenever complementary angles are involved.

Final Answer:
The value of cos²A + cos²B + cos²C is 1.