Let us complete the Pythagorean Identity:
csc2(x)−cot2(x)=(1sinx)2−(cosxsinx)2=1sin2x−cos2xsin2x=1−cos2xsin2x=sin2xsin2x=1.\csc^2(x)−\cot^2(x)=(\frac{1}{\sin x})^2-(\frac{\cos x}{\sin x})^2=\frac{1}{\sin^2 x}-\frac{\cos^2 x}{\sin^2 x}=\frac{1-\cos^2 x}{\sin^2 x}=\frac{\sin^2 x}{\sin^2 x}=1.csc2(x)−cot2(x)=(sinx1)2−(sinxcosx)2=sin2x1−sin2xcos2x=sin2x1−cos2x=sin2xsin2x=1.
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