Prove the identity sin(2x)/sin(x) - cos(2x)/cos(x)= sec x Can you explain steps please I'm having a hard time understanding this. Thanks in advance!

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

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Usually it's easiest to work with the more complicated side and try to reduce it to the simpler looking side. We'll use the identities `sin(2x)=2sin(x)cos(x)` and `cos(2x)=2cos^2(x)-1.` The left side then cancels pretty nicely as follows:

`(sin(2x))/sin(x)-(cos(2x))/cos(x)=(2sin(x)cos(x))/sin(x)-(2cos^2(x)-1)/cos(x)`

`=2cos(x)-((2cos^2(x))/cos(x)-1/cos(x))=2cos(x)-2cos(x)+1/cos(x)=1/cos(x)`

`=sec(x).`

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