Inverse Trig Functions

  1. How do you write each of the following as an algebraic function of xx? sin⁡(arctan⁡(x))sec⁡(arcsin⁡(x)) \sin\left(\arctan(x)\right) \qquad \qquad \sec\left(\arcsin(x)\right)

  2. Demonstrate how to calculate these algebraic formulas for the derivatives of inverse trigonometric functions.

    ddxarcsin⁡(x)=11−x2ddxarccsc(x)=−1xx2−1 \frac{\mathrm{d}}{\mathrm{d}x} \arcsin(x) = \frac{1}{\sqrt{1-x^2}} \qquad \frac{\mathrm{d}}{\mathrm{d}x} \mathrm{arccsc}(x) = -\frac{1}{x\sqrt{x^2-1}} ddxarccos⁡(x)=−11−x2ddxarcsec(x)=1xx2−1 \frac{\mathrm{d}}{\mathrm{d}x} \arccos(x) = -\frac{1}{\sqrt{1-x^2}} \qquad \frac{\mathrm{d}}{\mathrm{d}x} \mathrm{arcsec}(x) = \frac{1}{x\sqrt{x^2-1}} ddxarctan⁡(x)=11+x2ddxarccot(x)=−11+x2 \frac{\mathrm{d}}{\mathrm{d}x} \arctan(x) = \frac{1}{1+x^2} \qquad \frac{\mathrm{d}}{\mathrm{d}x} \mathrm{arccot}(x) = -\frac{1}{1+x^2}