`arccos(x) = arcsec(x)` Solve the equation for x

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Borys Shumyatskiy eNotes educator| Certified Educator

The function at the left side, `arccos(x),` is defined at `[-1, 1],` because it is the range of `cos(x).`  The function at the right side, `arcsec(x),` is defined at `(-oo, -1] uu [1, +oo],` because it is the range of `sec(x).`

Thus there are only two points at which both sides are defined, `x_1 = -1` and `x_2 = 1.` Let's check whether they are the solutions: `arccos(-1) = pi` and `arcsec(-1) = pi,` `arccos(1) = 0` and `arcsec(1) = 0.`

Yes, they are the solutions. The answer: `x_1 = -1,` `x_2 = 1.`

 

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