Question 1078896
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Solve the following equation, giving the exact solutions which lie in [0, 
tan^2(x) = 3/2 sec (x)
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{{{tan^2(x)}}} = {{{(3/2)*sec (x)}}}  <---->

{{{sin^2(x)/cos^2(x)}}} = {{{(3/2)*(1/cos(x))}}}  ---> multiply both sides by {{{cos^2(x)}}}. You will get

{{{sin^2(x)}}} = {{{(3/2)*cos(x)}}}  --->

{{{1-cos^2(x)}}} = {{{(3/2)*cos(x)}}}  --->

{{{2-2cos^2(x)}}} = {{{3*cos(x)}}}  --->

{{{2cos^2(x) + 3*cos(x) - 2}}} = 0  --->

{{{(2cos(x)-1)*(cos(x)+2)}}} = 0  ---> 


The last equation deploys in two independent equations


1.  2cos(x) - 1 = 0  --->  cos(x) = {{{1/2}}}  --->  x = {{{pi/3}}}  and/or  x = {{{5pi/3}}}.


2.  cos(x) + 2 = 0 --->  cos(x) = -2  ---  This equation has no solutions.


<U>Answer</U>.  The given equation has two roots  x = {{{pi/3}}}  and  x = {{{5pi/3}}} in the interval [{{{0}}},{{{2pi}}}).
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