Question

Find all solutions of each equation on the interval [0, 2pi]

1=cot^2x + csc x

Can anyone help me with this one? I'm stuck! :/

Answer 1

\[\csc ^{2}x-\cot ^{2}x=1,\cot ^{2}x=\csc ^{2}x-1\]
\[1=\csc ^{2}x-1+\csc x,\csc ^{2}x+\csc x-2=0\]
\[\csc ^{2}x+2\csc x-\csc x-2=0\]
\[\csc x \left( \csc x-2 \right)-1\left( \csc x-2 \right)=0\]
\[\left( \csc x-2 \right)\left( \csc x-1 \right)=0\]
\[either ~\csc x-2=0,\csc x=2,\sin x=\frac{ 1 }{ 2 }=\sin \frac{ \pi }{ 6 },\sin \left( \pi-\frac{ \pi }{6 } \right)\]
\[\sin x=\sin \frac{ \pi }{6 },\sin \frac{ 5\pi }{6 },x=\frac{ \pi }{ 6 },\frac{ 5\pi }{ 6 }\]
\[or~\csc x-1=1,\csc x=1,\sin x=1=\sin \frac{ \pi }{2 },x=\frac{ \pi }{ 2 }\]