HSF.TF.C.8 math practice. Learn by doing.

One form of the double-angle identity is $\cos(2\theta) = \cos^{2}\theta - \sin^{2}\theta\textsf{.}$

The fundamental Pythagorean identity is $\sin^{2}\theta + \cos^{2}\theta = 1\textsf{.}$

$\tan\theta = \dfrac{\sin\theta}{\cos\theta} = \dfrac{3/5}{4/5} = \dfrac{3}{4}\textsf{.}$

$\sec\theta = \dfrac{1}{\cos\theta} = \dfrac{1}{2/3} = \dfrac{3}{2}\textsf{.}$

$\sin(2\theta) = 2\sin\theta\cos\theta = 2 \cdot \dfrac{5}{13} \cdot \dfrac{12}{13} = \dfrac{120}{169}\textsf{.}$

The Pythagorean identity is $1 + \tan^{2}\theta = \sec^{2}\theta\textsf{.}$

$\tan\theta = \dfrac{\sin\theta}{\cos\theta} = \dfrac{-5/13}{12/13} = -\dfrac{5}{12}\textsf{.}$

$\sec\theta = \dfrac{1}{\cos\theta} = \dfrac{1}{-4/7} = -\dfrac{7}{4}\textsf{.}$