Chapter 5 - Section 5.5 - Additional Identities - 5.5 Problem Set - Page 312: 55

$d$

$\theta = \tan^{-1} {x} $ $\tan{\theta} = x$ $\sec{\theta} = \sqrt{1+\tan^2{\theta}} = \sqrt{1+x^2}$ $\cos{\theta} =\dfrac{1}{\sec{\theta}} = \dfrac{1}{\sqrt{1+x^2}}$ $\sin{\theta} = \sqrt{1-\cos^2{\theta}} = \dfrac{x}{\sqrt{1+x^2}}$ $\sin{2\theta} = 2 \sin{\theta} \cos{\theta} = \dfrac{2x}{1+x^2}$