If x = a cos θ cos φ, y = a cos θ sin φ and z = a sin θ, then what is the simplified value of x² + y² + z² in terms of a?

Introduction / Context:
This question comes from coordinate geometry and trigonometry, where variables are expressed in terms of angles and a common parameter a. The expression x² + y² + z² often represents the square of a distance in three dimensional space. The task is to simplify this sum using trigonometric identities.

Given Data / Assumptions:

• x = a cos θ cos φ.
• y = a cos θ sin φ.
• z = a sin θ.
• We want to find x² + y² + z² in terms of a only.

Concept / Approach:
Important ideas:

• Square each of x, y and z and add the results.
• Use the identity cos²φ + sin²φ = 1.
• Use the identity sin²θ + cos²θ = 1.
• Factor out common constants and terms like a² to simplify.

Step-by-Step Solution:
Step 1: Compute x²: x² = (a cos θ cos φ)² = a² cos²θ cos²φ. Step 2: Compute y²: y² = (a cos θ sin φ)² = a² cos²θ sin²φ. Step 3: Compute z²: z² = (a sin θ)² = a² sin²θ. Step 4: Add them: x² + y² + z² = a² cos²θ cos²φ + a² cos²θ sin²φ + a² sin²θ. Step 5: Factor out a²: x² + y² + z² = a²[cos²θ cos²φ + cos²θ sin²φ + sin²θ]. Step 6: Inside the bracket, factor cos²θ: cos²θ(cos²φ + sin²φ) + sin²θ. Step 7: Use cos²φ + sin²φ = 1, giving cos²θ × 1 + sin²θ = cos²θ + sin²θ. Step 8: Apply sin²θ + cos²θ = 1 to get x² + y² + z² = a² × 1 = a².
Verification / Alternative check:
Step 1: Choose specific angles, for example θ = 0 and φ = 0, and a = 1. Step 2: Then x = 1 × cos 0 × cos 0 = 1, y = 1 × cos 0 × sin 0 = 0, z = 1 × sin 0 = 0. Step 3: Compute x² + y² + z² = 1² + 0² + 0² = 1, which equals a² when a = 1. Step 4: Trying other values of θ and φ numerically will also give x² + y² + z² = a², reinforcing the algebraic result.
Why Other Options Are Wrong:
Option 2a² or 4a² would require the sum inside the bracket to be 2 or 4 instead of 1, which contradicts the identities used. Option 0 would imply x² + y² + z² is always zero, which is impossible unless a is zero. Option a has incorrect units, since squaring variables produces a quantity proportional to a², not a.
Common Pitfalls:
Forgetting to factor out a² early often leads to more complicated algebra than necessary. Some learners misapply cos²φ + sin²φ = 1 and try to combine it incorrectly with θ instead of φ. It is also common to overlook that cos²θ + sin²θ = 1 regardless of θ, as long as θ is a real angle.
Final Answer:
The simplified value of x² + y² + z² is a².