`sqrt(1+sinx)/sqrt(1-sinx)` =secx(secx+tanx)

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Solve `sqrt(1+sinx)/sqrt(1-sinx)=secx(secx+tanx)`


`sqrt(1+sinx)/sqrt(1-sinx) * sqrt(1+sinx)/sqrt(1+sinx)=secx(secx+tanx)`




(a) For cosx>0:



If secx+tanx=0 then secx=-tanx ==> sinx=-1 which is not in the domain of the original problem, so `secx+tanx!=0` .

Then secx=1 (after dividing both sides by secx+tanx) and `x=2npi,n in ZZ` (n an integer).

(b) For cosx<0:

|cosx|=-cosx so:


Again `secx+tanx!=0` so dividing we get

secx=-1 ==> `x=pi+2npi,n in ZZ`


The solutions are `x=npi,n in Z`


The graph of the left side in black, the right side in red: