Question 1086310
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First thing you need to know and to apply is  {{{tan(x-pi/2)}}} = cot(x) = {{{cos(x)/sin(x)}}}.


When you apply it, your equation becomes


{{{10*(cos(x)/sin(x))*sin^2(x)}}} = {{{5*(sqrt(3)/2)}}},   or

{{{2*cos(x)*sin(x)}}} = {{{sqrt(3)/2}}}.


Second thing you need to know is  2*cos(x)*sin(x) = sin(2x).


When you apply it, the last equation becomes


sin(2x) = {{{sqrt(3)/2}}}.


It has the solutions  2x = {{{pi/3}}},  2x = {{{2pi/3}}},  2x = {{{pi/3 + 2pi}}}  and  2x = {{{(2pi)/3 + 2pi}}}.


Therefore, the solutions for x are  {{{pi/6}}},  {{{pi/3}}},  {{{7pi/6}}}  and  {{{(4pi)/3}}}.
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Solved.