This is the chapter with the most demonstration exercises. Before opening a solution, decide which family of formulas to use: addition, multiplication, division or transformation into a product. Addition formulas 131 Calculate the values of: a) $\cos 15^\circ$; b) $\operatorname{sen} 105^\circ$; c) $\operatorname{tg} 75^\circ$; d) $\sec 285^\circ$. See the resolutiona) $\cos 15^\circ = \cos(45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \operatorname{sen} 45^\circ \operatorname{sen} 30^\circ = \dfrac{\sqrt{2}}{2} \cdot \dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2} \cdot \dfrac{1}{2} = \dfrac{\sqrt{6} + \sqrt{2}}{4}$ b) $\operatorname{sen} 105^\circ = \operatorname{sen}(60^\circ + 45^\circ) = \operatorname{sen} 60^\circ \cos 45^\circ + \operatorname{sen} 45^\circ \cos 60^\circ = \dfrac{\sqrt{3}}{2} \cdot \dfrac{\sqrt{2}}{2} + \dfrac{\sqrt{2}}{2} \cdot \dfrac{1}{2} = \dfrac{\sqrt{6} + \sqrt{2}}{4}$ c) $\operatorname{tg} 75^\circ = \operatorname{tg}(45^\circ + 30^\circ) = \dfrac{\operatorname{tg} 45^\circ + \operatorname{tg} 30^\circ}{1 - \operatorname{tg} 45^\circ \operatorname{tg} 30^\circ} = \dfrac{1 + \frac{\sqrt{3}}{3}}{1 - \frac{\sqrt{3}}{3}} = \dfrac{3 + \sqrt{3}}{3 - \sqrt{3}}$ $\left(= 2 + \sqrt{3}\right)$ d) $\sec 285^\circ = \sec 75^\circ = \dfrac{1}{\cos 75^\circ} = \dfrac{1}{\cos(45^\circ + 30^\circ)} = \dfrac{4}{\sqrt{6} - \sqrt{2}}$ $\left(= \sqrt{6} + \sqrt{2}\right)$
In elementary algebra, transforming a polynomial into a product (factoring) is a resource of enormous practical importance: $x^2 - 2x = x(x - 2)$ highlighting $x^2 - 4 = (x + 2)(x - 2)$ difference of squares $x^2 \pm 4x + 4 = (x \pm 2)^2$ perfect square trinomials $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$ sum of cubes $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$ difference of cubes $x^3 \pm 3x^2 + 3x \pm 1 = (x \pm 1)^3$ perfect cubes We will often apply these features to trigonometry, with transformations like $\operatorname{sen}^2 x - 2\operatorname{sen} x = \operatorname{sen} x(\operatorname{sen} x - 2)$ and $\operatorname{sen}^2 x - \cos^2 x = (\operatorname{sen} x + \cos x)(\operatorname{sen} x - \cos x)$. In addition to algebraic resources, trigonometry has formulas that allow you to complete a factorization — in the example above, also factor $\operatorname{sen} x + \cos x$ and $\operatorname{sen} x - \cos x$.