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HomertheGenius

\(\frac{cot^{2} x}{1 + csc x}+ sin x * csc x = \\ = \frac{ \frac{cos ^{2} x}{sin ^{2} x} }{1+ \frac{1}{sin x} }+ sin x * \frac{1}{sin x}= \\ = \frac{ \frac{cos^{2} x}{sin ^{2} x} }{ \frac{sin x + 1}{sin x} }+ 1 = \\ = \frac{cos ^{2} x}{sin x ( sin x + 1 )}+ 1 = \\ = \frac{cos ^{2} x+sin x ( sin x + 1 )}{sin x ( sin x + 1 )}= \\ = \frac{cos ^{2} x+sin ^{2}x+ sin x }{sin x( sin x + )}= \\ = \frac{sinx+1}{sinx( sinx+1)} = \frac{1}{sin x}= csc x\)

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