[tex]\bf 1+cot^2(\theta)=csc^2(\theta)\implies cot^2(\theta)=csc^2(\theta)-1
\\\\\\
csc(\theta)=\cfrac{1}{sin(\theta)}
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cot^2(x)+csc(x)=1\implies csc^2(x)-1+csc(x)=1
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csc^2(x)+csc(x)-2=0\implies [csc(x)-2][csc(x)+1]=0\\\\
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[tex]\bf csc(x)-2=0\implies csc(x)=2\implies \cfrac{1}{sin(x)}=2\implies \cfrac{1}{2}=sin(x)
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sin^{-1}\left( \frac{1}{2} \right)=\measuredangle x\implies \frac{\pi }{6}~~,~~\frac{5\pi }{6}=\measuredangle x\\\\
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csc(x)+1=0\implies csc(x)=-1\implies \cfrac{1}{sin(x)}=-1
\\\\\\
\cfrac{1}{-1}=sin(x)
\implies
-1=sin(x)\implies sin^{-1}(-1)=\measuredangle x\implies \frac{3\pi }{2}=\measuredangle x[/tex]
