Asked by PRINCE
Prove that:sin^2x (1+cot^2x)=1
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GPT-5 mini
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Use cot x = cos x / sin x (so require sin x ≠ 0). Then
sin^2 x(1 + cot^2 x) = sin^2 x + sin^2 x·cot^2 x
= sin^2 x + sin^2 x·(cos^2 x / sin^2 x)
= sin^2 x + cos^2 x
= 1,
by the Pythagorean identity sin^2 x + cos^2 x = 1. (Equivalently, 1 + cot^2 x = csc^2 x = 1/sin^2 x, so sin^2 x·csc^2 x = 1.) The identity holds for all x where cot x (or csc x) is defined, i.e. sin x ≠ 0.
sin^2 x(1 + cot^2 x) = sin^2 x + sin^2 x·cot^2 x
= sin^2 x + sin^2 x·(cos^2 x / sin^2 x)
= sin^2 x + cos^2 x
= 1,
by the Pythagorean identity sin^2 x + cos^2 x = 1. (Equivalently, 1 + cot^2 x = csc^2 x = 1/sin^2 x, so sin^2 x·csc^2 x = 1.) The identity holds for all x where cot x (or csc x) is defined, i.e. sin x ≠ 0.
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