Demostrar que $\frac{1+2\sin{\theta}\cos{\theta}}{\cos^2{\theta}-\sin^2{\theta}}=\frac{1+\tan{\theta}}{1-\tan{\theta}}$

Question

Demostrar que $$\frac{1+2\sin{\theta}\cos{\theta}}{\cos^2{\theta}-\sin^2{\theta}}=\frac{1+\tan{\theta}}{1-\tan{\theta}}$$

Este es mi intento $$\require{cancel}\text{Left - Right} = \frac{(1+2\sin{\theta}\cos{\theta})(1-\tan{\theta})-(1+\tan{\theta})(\cos^2{\theta}-\sin^2{\theta})}{(\cos^2{\theta}-\sin^2{\theta})(1-\tan{\theta})}= \frac{\cancel1\cancel{-\tan{\theta}}+2\sin^2{\theta}\cancel{+2\sin{\theta}\cos{\theta}}\cancel{-\cos^2{\theta}+\sin^2{\theta}}\cancel{-\sin{\theta}\cos{\theta}}\cancel{+\tan{\theta}}\cancel{-\sin{\theta}\cos{\theta}}}{(\cos^2{\theta}-\sin^2{\theta})(1-\tan{\theta})} =\frac{2\sin^2{\theta}}{(\cos^2{\theta}-\sin^2{\theta})(1-\tan{\theta})}$$ ¿Debería ser cero, pero no lo es? Aquí está la prueba: $$\text{Left}=\frac{\sin^2{\theta}+\cos^2{\theta}+2\sin{\theta}\cos{\theta}}{\cos^2{\theta}-\sin^2{\theta}}=\frac{(\sin{\theta}+\cos{\theta})^2}{(\cos{\theta}+\sin{\theta})(\cos{\theta}-\sin{\theta})}=\frac{\sin{\theta}+\cos{\theta}}{\cos{\theta}-\sin{\theta}} \\\text{Right}=\frac{1+\frac{\sin{\theta}}{\cos{\theta}}}{1-\frac{\sin{\theta}}{\cos{\theta}}}=\frac{\cos{\theta}+\sin{\theta}}{\cos{\theta}-\sin{\theta}} \\{\therefore}\text{Left=Right}$$

Answer 1

De forma directa:

Factor de salida $\cos^2\theta$ a partir del numerador y el denominador de la fracción l.h.s. y haz algo de trigonometría: $$\frac{1+2\sin\theta\cos\theta}{\cos^2\theta-\sin^2\theta}=\frac{\dfrac1{\cos^2\theta}+2\tan\theta}{1-\tan^2\theta}=\frac{1+\tan^2\theta+2\tan\theta}{1-\tan^2\theta}=\frac{(1+\tan\theta)^2}{1-\tan^2\theta}$$ y simplificar.

Answer 2

Tenga en cuenta que $$ 1+2\sin\theta\cos\theta=\cos^2\theta+\sin^2\theta+2\sin\theta\cos\theta= (\cos\theta+\sin\theta)^2 $$ Por lo tanto, \begin{align} \frac{1+2\sin\theta\cos\theta}{\cos^2\theta-\sin^2\theta} &=\frac{(\cos\theta+\sin\theta)^2}{(\cos\theta+\sin\theta)(\cos\theta-\sin\theta)} \\[6px] &=\frac{\cos\theta+\sin\theta}{\cos\theta-\sin\theta} \\[6px] &=\frac{\cos\theta(1+\tan\theta)}{\cos\theta(1-\tan\theta)} \\[6px] &=\frac{1+\tan\theta}{1-\tan\theta} \end{align}

Answer 3

Utilice \begin{equation} 2 \sin \theta \cos \theta = \sin 2\theta = \frac{2 \tan \theta}{1 + \tan^2 \theta} \end{equation} \begin{equation} \cos^2 \theta - \sin^2 \theta = \cos 2\theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \end{equation} Sustituyendo las cantidades anteriores, obtenemos \begin{equation} \frac{1 + \frac{2 \tan \theta}{1 + \tan^2 \theta}}{\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta}} = \frac{1 + \tan^2 \theta + 2 \tan \theta}{1 - \tan^2 \theta} = \frac{(1 + \tan \theta)^2}{( 1 - \tan \theta) (1 + \tan \theta)} = \frac{1 + \tan \theta}{1 - \tan \theta} \end{equation}

Answer 4

Abreviando $\sin\theta$ por $s$ y $\cos\theta$ por $s$ tenemos en la base de $c^2+s^2=1$ : $$1+2sc=c^2+2sc+s^2=(c+s)^2$$ para que: $$(1+2sc)(c-s)=(c+s)^2(c-s)=(c^2-s^2)(c+s)$$ Dividiendo ambos lados por $c$ y abreviando $\tan(\theta)$ por $t$ nos ponemos en la base de $t=s/c$ : $$(1+2sc)(1-t)=(c^2-s^2)(1+t)$$

Ahora divide ambos lados por $(1-t)(c^2-s^2)$ para conseguirlo: $$\frac{1+2sc}{c^2-s^2}=\frac{1+t}{1-t}$$

Answer 5

$\dfrac{1+\tan t}{1-\tan t} =$

$\dfrac{\cos t + \sin t}{\cos t -\sin t}=$

$\dfrac{(\cos t+\sin t)^2}{\cos^2 t - \sin^2t}=$

$\dfrac{1+2\sin t \cos t}{\cos^2 t-\sin^2 t}$ .