Aprende con Inteligencia

Resolver la ecuación:
$$ 4x^4 + x^6 = -\sin^2 5x $$

Paso 1:
Si se cumple que: $\frac{\sec x + a \tan x}{\sec x + a} = \frac{\sin x - a \tan x}{\sin x - a}$, halle: $\sec^2 x + \csc^2 x$

Si $\tan 25^{\circ} = a$, demostrar que:
$$ \frac{\tan 155^{\circ} - \tan 115^{\circ}}{1 + \tan 155^{\circ} \cdot \tan 115^{\circ}} = \frac{1 - a^{2}}{2a} $$

Paso 1:
Hallar el rango de la función $f(x) = \sin \left( \sqrt{\frac{\pi^2}{36} - x^2} \right)$.

Para un entero positivo $n$, sea:
$$ f_n(\theta) = \tan\left(\frac{\theta}{2}\right)(1 + \sec \theta)(1 + \sec 2\theta)(1 + \sec 4\theta)\dots(1 + \sec 2^n \theta) $$
Entonces:

(a) $f_2\left(\frac{\pi}{16}\right) = 1$      (b) $f_3\left(\frac{\pi}{32}\right) = 1$      (c) $f_4\left(\frac{\pi}{64}\right) = 1$      (d) $f_5\left(\frac{\pi}{128}\right) = 1$

Resolver:
$$ \begin{cases} \sin y = 5 \sin x \\ 3 \cos x + \cos y = 2 \end{cases} $$

Demuestre la siguiente identidad trigonométrica:
$$ \frac{\cot \theta - \tan \theta}{1 - 2\sin^2 \theta} = \sec \theta \csc \theta $$

Resolver la ecuación:
$$\tan(x - 45^\circ) \tan x \tan(x + 45^\circ) = \frac{4 \cos^2 x}{\tan \frac{x}{2} - \cot \frac{x}{2}}$$

Demostrar que:
$$ \arcsin \frac{4}{5} + \arcsin \frac{5}{13} + \arcsin \frac{16}{65} = \frac{\pi}{2} $$

Demuestre que:
$$ \tan x \cdot \tan 2x + \tan 2x \cdot \tan 3x + \dots + \tan nx \cdot \tan(n+1)x = \cot x [\tan(n+1)x - \tan x] - n $$

Resolver la ecuación:
$$ \tan x + \cot x - \cos 4x = 3 $$

Resolver el sistema:
$$ \begin{cases} \sin x = 3 \sin y \\ \tan x = 5 \tan y \end{cases} $$