$$ \left. \begin{array} { l } { 1 \cdot \frac { \cot heta + 2 \cos heta } { \cos heta - \sin heta } = \sec heta + 2 an heta } \ { herefore m \frac { \cot heta } { \cos heta } + \frac { \sec heta } { \cot heta } = \sec ^ { 2 } heta \csc heta } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { 1 \cdot \frac { \cot 	heta + 2 \cos 	heta } { \cos 	heta - \sin 	heta } = \sec 	heta + 2 	an 	heta } \ { 	herefore m \frac { \cot 	heta } { \cos 	heta } + \frac { \sec 	heta } { \cot 	heta } = \sec ^ { 2 } 	heta \csc 	heta } \end{array} ight. $$

UpStudy AI · Accepted Answer

$$
\textbf{Ecuación 2:} \quad m\,\frac{\cot \theta}{\cos \theta} + \frac{\sec \theta}{\cot \theta} = \sec^2 \theta\,\csc \theta
$$

1. Escribimos las funciones en términos de seno y coseno.

$$
\cot \theta = \frac{\cos \theta}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta}
$$

2. Simplifiquemos el primer término:

$$
m\,\frac{\cot \theta}{\cos \theta} = m\,\frac{\frac{\cos \theta}{\sin \theta}}{\cos \theta} = \frac{m}{\sin \theta}
$$

3. Simplifiquemos el segundo término:

$$
\frac{\sec \theta}{\cot \theta} = \frac{\frac{1}{\cos \theta}}{\frac{\cos \theta}{\sin \theta}} = \frac{1}{\cos \theta}\cdot\frac{\sin \theta}{\cos \theta} = \frac{\sin \theta}{\cos^2 \theta}
$$

4. Escribamos el lado derecho:

$$
\sec^2 \theta\,\csc \theta = \frac{1}{\cos^2 \theta} \cdot \frac{1}{\sin \theta} = \frac{1}{\cos^2 \theta\,\sin \theta}
$$

5. La ecuación se transforma en:

$$
\frac{m}{\sin \theta} + \frac{\sin \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta\,\sin \theta}
$$

6. Multiplicamos ambos lados de la ecuación por $$\cos^2 \theta\,\sin \theta$$ para eliminar denominadores:

$$
m\,\cos^2 \theta\,\sin \theta\cdot\frac{1}{\sin \theta} + \frac{\sin \theta}{\cos^2 \theta}\cdot\cos^2 \theta\,\sin \theta = \frac{1}{\cos^2 \theta\,\sin \theta}\cdot \cos^2 \theta\,\sin \theta
$$

Simplificando cada término:

$$
m\,\cos^2 \theta + \sin^2 \theta = 1
$$

7. Despejamos $$m$$:

$$
m\,\cos^2 \theta = 1 - \sin^2 \theta
$$

Utilizamos la identidad pitagórica $$\sin^2 \theta + \cos^2 \theta = 1$$ para notar que:

$$
1 - \sin^2 \theta = \cos^2 \theta
$$

Luego:

$$
m\,\cos^2 \theta = \cos^2 \theta
$$

Siempre que $$\cos^2 \theta \neq 0$$, se tiene:

$$
m = 1
$$

$$
\boxed{m = 1}
$$
$$ m = 1 $$

\( \left. \begin{array} { l } { 1 \cdot \frac { \cot \theta + 2 \cos \theta } { \cos \theta - \sin \theta } = \sec \theta + 2 \tan \theta } \\ { \therefore m \frac { \cot \theta } { \cos \theta } + \frac { \sec \theta } { \cot \theta } = \sec ^ { 2 } \theta \csc \theta } \end{array} \right. \)

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