If x = p secA cosB, y = q secA sinB and z = r tanA, what is the value of the expression (x² / p²) + (y² / q²) - (z² / r²)?

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Given:  
 x = p secA cosB  
 y = q secA sinB  
 z = r tanA

We need to find the value of (x² / p²) + (y² / q²) - (z² / r²).

Step 1: Calculate each term separately.

(x² / p²) = (p² sec²A cos²B) / p² = sec²A cos²B

(y² / q²) = (q² sec²A sin²B) / q² = sec²A sin²B

(z² / r²) = (r² tan²A) / r² = tan²A

Step 2: Sum the first two terms:  
sec²A cos²B + sec²A sin²B = sec²A (cos²B + sin²B) = sec²A * 1 = sec²A

Step 3: Now the expression becomes:  
sec²A - tan²A

Step 4: Use the trigonometric identity:  
sec²A - tan²A = 1

Therefore, the value of the expression is 1.

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