Question 747134
{{{cot(x)=cos(x)/sin(x)}}} and {{{csc(x)=1/sin(x)}}} so
{{{(cot(x))^2+(csc(x))^2=7}}} --> {{{(cos(x))^2/(sin(x))^2+1/(sin(x))^2=7}}}
Multiplying both sides times {{{(sin(x))^2}}} we can eliminate denominators to get
{{{(cos(x))^2+1=7(sin(x))^2}}} --> {{{1-(sin(x))^2+1=7(sin(x))^2}}} --> {{{2-(sin(x))^2=7(sin(x))^2}}} --> {{{2=8(sin(x))^2}}} --> {{{2/8=(sin(x))^2}}} --> {{{(sin(x))^2=1/4}}}
The solutions must yield {{{sin(x)=1/2}}} or {{{sin(x)=-1/2}}}
Looking for solution between {{{0^o}}} and {{{360^o}}} (between {{{0}}} and {{{2pi}}} radians),
{{{sin(x)=1/2}}} happens for
{{{x=30^o}}} ({{{pi/6}}}radians) and {{{x=150^o}}} ({{{5pi/6}}}radians)
and {{{sin(x)=-1/2}}} happens for
{{{x=210^o}}} ({{{7pi/6}}}radians) and {{{x=330^o}}} ({{{11pi/6}}}radians).
All solutions can be expressed as 
{{{x=k*180^o +- 30^o}}} or {{{x=k*pi +- pi/6}}}