Question

cos^(-1)⁡(-x)= - cos^(-1)⁡(x)or cos^(-1)⁡(x) ?? which one is correct?

sec^(-1)⁡(-x)= - sec^(-1)⁡(x)or sec^(-1)⁡(x) ?? which one is correct?

Answer 1

Answer for Question 50664, Math, Trigonometry:

$$\cos^{-1}(-x) = \arccos(-x) = \pi - \arccos(x) = \pi - \cos^{-1}(x).$$ $$\sec^{-1}(-x) = \arcsec(-x) = \arccos\left(-\frac{1}{x}\right) = \cos^{-1}\left(-\frac{1}{x}\right).$$ $$\arcsec(-x) = A;$$ $$- x = \sec A;$$ $$x = -\sec A;$$ $$x = -\frac{1}{\cos A};$$ $$\cos A = -\frac{1}{x};$$ $$A = \arccos\left(-\frac{1}{x}\right);$$ $$\arcsec(-x) = \arccos\left(-\frac{1}{x}\right).$$

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