Demostración de igualdades trigonométricas \( \sin^2(x) + \cos^2(x) = 1 \) \( \frac{\sin(4x) + \sin(2x)}{\cos(4x) + \cos(2x)} = \tan(3x) \) \( \tan^2(x) = \frac{\sin^2(x)}{1 – \sin^2(x)} \) \( \sec(x) = \sin(x)\bigl(\tan(x) + \cot(x)\bigr) \) \( \sin(2x) = \frac{2\,\tan(x)}{1 + \tan^2(x)} \) \( \frac{2\,\sin(x) + 3}{2\,\tan(x) + 3\,\sec(x)} = \cos(x) \) \( \frac{1 – \sin^4(x)}{\cos^2(x)} = 2 – \cos^2(x) \) \( \frac{2\,\sin(x)}{\tan(2x)} = \cos(x) – \frac{\sin^2(x)}{\cos(x)} \) \( \frac{\cos^2(\alpha)}{1 + \sin(\alpha)} = 1 – \sin(\alpha) \) \( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \)