English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

verificar 2csc(2x)=sec^2(x)cot(x)

Solución

verificar

2 csc (2 x) = sec^{2} (x) cot (x)

Verdadero

Pasos de solución

2 csc (2 x) = sec^{2} (x) cot (x)

Manipular el lado derecho

2 csc (2 x)

Expresar con seno, coseno

2 csc (2 x)

Utilizar la identidad trigonométrica básica:

csc (x) = \frac{1}{sin ( x )}

= 2 \cdot \frac{1}{sin ( 2 x )}

Simplificar

2 \cdot \frac{1}{sin ( 2 x )} : \frac{2}{sin ( 2 x )}

2 \cdot \frac{1}{sin ( 2 x )}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{sin ( 2 x )}

Multiplicar los numeros:

1 \cdot 2 = 2

= \frac{2}{sin ( 2 x )}

= \frac{2}{sin ( 2 x )}

= \frac{2}{sin ( 2 x )}

Re-escribir usando identidades trigonométricas

\frac{2}{sin ( 2 x )}

Utilizar la identidad trigonométrica del ángulo doble:

sin (2 x) = 2 sin (x) cos (x)

= \frac{2}{2 sin ( x ) cos ( x )}

Dividir:

\frac{2}{2} = 1

= \frac{1}{sin ( x ) cos ( x )}

= \frac{1}{sin ( x ) cos ( x )}

Manipular el lado izquierdo

sec^{2} (x) cot (x)

Expresar con seno, coseno

cot (x) sec^{2} (x)

Utilizar la identidad trigonométrica básica:

cot (x) = \frac{cos ( x )}{sin ( x )}

= \frac{cos ( x )}{sin ( x )} sec^{2} (x)

Utilizar la identidad trigonométrica básica:

sec (x) = \frac{1}{cos ( x )}

= \frac{cos ( x )}{sin ( x )} (\frac{1}{cos ( x )})^{2}

Simplificar

\frac{cos ( x )}{sin ( x )} (\frac{1}{cos ( x )})^{2} : \frac{1}{cos ( x ) sin ( x )}

\frac{cos ( x )}{sin ( x )} (\frac{1}{cos ( x )})^{2}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{cos ( x ) ( \frac{1}{c o s ( x )} ) ^{2}}{sin ( x )}

(\frac{1}{cos ( x )})^{2} = \frac{1}{cos ^{2} ( x )}

(\frac{1}{cos ( x )})^{2}

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{cos ^{2} ( x )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( x )}

= \frac{\frac{1}{c o s ^{2} ( x )} cos ( x )}{sin ( x )}

Multiplicar

cos (x) \frac{1}{cos ^{2} ( x )} : \frac{1}{cos ( x )}

cos (x) \frac{1}{cos ^{2} ( x )}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot cos ( x )}{cos ^{2} ( x )}

Multiplicar:

1 \cdot cos (x) = cos (x)

= \frac{cos ( x )}{cos ^{2} ( x )}

Eliminar los terminos comunes:

cos (x)

= \frac{1}{cos ( x )}

= \frac{\frac{1}{c o s ( x )}}{sin ( x )}

Aplicar las propiedades de las fracciones:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{1}{cos ( x ) sin ( x )}

= \frac{1}{cos ( x ) sin ( x )}

= \frac{1}{cos ( x ) sin ( x )}

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

cos^{2} (x) - cos^{2} (x) = 0

2 (sec (x) - tan (x)) (csc (x) + 1) = 2 cot (x)

sec (x) sin (x) cot (x) = 1

\frac{cos ^{2} ( x )}{cot ( x )} = sin (x) cos (x)

cos^{2} (α) sec^{2} (α) = 1