If \tan x-\sin x=a, what is \sec^3(x)-1 in terms of

Averie Ray

Averie Ray

Answered question

2022-04-22

If tanxsinx=a, what is sec3(x)1 in terms of a

Answer & Explanation

Yaretzi Odom

Yaretzi Odom

Beginner2022-04-23Added 16 answers

We can recall the difference of cubes formula
a3b3=(ab)(a2+2ab+b2)
So now we see that
sec3x1=(secx1)(sec2x2secx+1)
Or that
sec3x1=(Asinx)(sec2x2secx+1)
Continue with the simplification process.
sec3x1=Asinx(22secx+tan2x)
sec3x1=Asinx(Asinx+tan2x)
sec3x1=Asinxcos2xA2sin2x
We can't go much further than that, unfortunately. I would like to see if any contributors have other ideas.

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