SOLUTION: Find all solutions in the interval [0, 2π). sec^2 x - 2 = tan^2 x

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Question 1142428: Find all solutions in the interval [0, 2π).
sec^2 x - 2 = tan^2 x
Found 2 solutions by math_helper, MathTherapy:
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!


+sec%5E2%28x%29+-+2+=+tan%5E2%28x%29+

+%281%2Fcos%5E2%28x%29%29+-+2+=+sin%5E2%28x%29%2Fcos%5E2%28x%29+

++1+-+2cos%5E2%28x%29+=+sin%5E2%28x%29+

++%28+1-cos%5E2%28x%29+%29+-+cos%5E2%28x%29+=+sin%5E2%28x%29+ 

++sin%5E2%28x%29++-+cos%5E2%28x%29+=+sin%5E2%28x%29+



Now we see that we are looking for +-cos%5E2%28x%29+=+0+ or just +cos%28x%29+=+0+ on  [ +0+, +2pi+ )



+cos%28x%29+=+0+ at  +pi%2F2++ and ++3pi%2F2++ only





But +cos%5E2%28x%29=0+ implies the original equation is undefined (look at the 2nd line from the top), thus there is +highlight%28matrix%281%2C2%2CNO%2C+SOLUTION%29%29+.



Graphing we can see this, the zero crossings for cos(x) line up with undefined values for the other two functions:







+green%28tan%5E2%28x%29%29+,  +red%28sec%5E2%28x%29-2%29+, blue%28cos%28x%29%29+

Answer by MathTherapy(10557) About Me  (Show Source):
You can put this solution on YOUR website!

Find all solutions in the interval [0, 2π).
sec^2 x - 2 = tan^2 x
matrix%281%2C3%2C+sec%5E2+%28x%29+-+2%2C+%22=%22%2C+tan%5E2+%28x%29%29

matrix%281%2C3%2C+sec%5E2+%28x%29+-+2%2C+%22=%22%2C+sec%5E2+%28x%29+-+1%29 ----- Substituting sec%5E2+%28x%29+-+1 for tan%5E2+%28x%29

matrix%281%2C3%2C+sec%5E2+%28x%29+-+sec%5E2+%28x%29+-+2%2C+%22=%22%2C+-+1%29 =====>