SOLUTION: Solve the following equation, giving the exact solutions which lie in [0, tan^2(x) = 3/2 sec (x)

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Question 1078896: Solve the following equation, giving the exact solutions which lie in [0,
tan^2(x) = 3/2 sec (x)
Answer by ikleyn(52897) About Me  (Show Source):
You can put this solution on YOUR website!
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Solve the following equation, giving the exact solutions which lie in [0,
tan^2(x) = 3/2 sec (x)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

tan%5E2%28x%29 = %283%2F2%29%2Asec+%28x%29  <---->

sin%5E2%28x%29%2Fcos%5E2%28x%29 = %283%2F2%29%2A%281%2Fcos%28x%29%29  ---> multiply both sides by cos%5E2%28x%29. You will get

sin%5E2%28x%29 = %283%2F2%29%2Acos%28x%29  --->

1-cos%5E2%28x%29 = %283%2F2%29%2Acos%28x%29  --->

2-2cos%5E2%28x%29 = 3%2Acos%28x%29  --->

2cos%5E2%28x%29+%2B+3%2Acos%28x%29+-+2 = 0  --->

%282cos%28x%29-1%29%2A%28cos%28x%29%2B2%29 = 0  ---> 


The last equation deploys in two independent equations


1.  2cos(x) - 1 = 0  --->  cos(x) = 1%2F2  --->  x = pi%2F3  and/or  x = 5pi%2F3.


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


Answer.  The given equation has two roots  x = pi%2F3  and  x = 5pi%2F3 in the interval [0,2pi).