If cotA = P/Q and A is an acute angle, find the value of 1/2 (P secA + Q cosecA)².
2(P² + Q²)
Given cotA = P/Q with A acute, treat P and Q as the adjacent and opposite sides of a right triangle, so the hypotenuse is √(P² + Q²). Then sinA = Q/√(P² + Q²) and cosA = P/√(P² + Q²).
Therefore secA = √(P² + Q²)/P and cosecA = √(P² + Q²)/Q.
P·secA + Q·cosecA = P · √(P² + Q²)/P + Q · √(P² + Q²)/Q = √(P² + Q²) + √(P² + Q²) = 2√(P² + Q²).
Squaring, (P·secA + Q·cosecA)² = 4(P² + Q²), and halving gives 1/2 × 4(P² + Q²) = 2(P² + Q²). Hence, the correct answer is 2(P² + Q²).
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