Question Detail

(sin x + cos x)² = sin²x + cos²x + 2 sin x cos x = 1 + 2 sin x cos x. (csc x - cot x)² = (1/sinx - cosx/sinx)² = ((1-cosx)/sinx)² = (1-cosx)²/sin²x. Using identity (1-cosx)/sinx = tan(x/2), we get tan²(x/2). But the difference is not obviously 2. Let's test with x=45°: sin45=cos45=√2/2, so (sin+cos)² = (√2)² = 2. csc45=√2, cot45=1, so (√2-1)² = 2 - 2√2 + 1 = 3 - 2√2 ≈ 0.17. Difference ≈ 1.83, not 2. Test x=30°: sin30=1/2, cos30=√3/2, sum= (1+√3)/2, square = (1+3+2√3)/4 = (4+2√3)/4 = 1 + √3/2 ≈ 1.866. csc30=2, cot30=√3, (2-√3)² = 4 - 4√3 + 3 = 7 - 4√3 ≈ 0.072. Difference ≈ 1.794. Not 2 consistently.