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

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embizze | High School Teacher | (Level 1) Educator Emeritus

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

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

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

`(1+sinx)/sqrt(1-sin^2x)=secx(secx+tanx)`

`(1+sinx)/sqrt(cos^2x)=secx(secx+tanx)`

`(1+sinx)/|cosx|=secx(secx+tanx)`

(a) For cosx>0:

|cosx|=cosx

secx+tanx=secx(secx+tanx)

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:

-secx-tanx=secx(secx+tanx)

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

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

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

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

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