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verificar cos(x)+sin(x)tan(x)=sec(x)

Solución

verificar

cos (x) + sin (x) tan (x) = sec (x)

Verdadero

Pasos de solución

cos (x) + sin (x) tan (x) = sec (x)

Manipular el lado derecho

cos (x) + sin (x) tan (x)

Expresar con seno, coseno

cos (x) + sin (x) tan (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

cos (x) + sin (x) \frac{sin ( x )}{cos ( x )}

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

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

Multiplicar fracciones:

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

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

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

Sumar:

1 + 1 = 2

= sin^{2} (x)

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

cos (x) cos (x) + sin^{2} (x) = cos^{2} (x) + sin^{2} (x)

cos (x) cos (x) + sin^{2} (x)

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

cos (x) cos (x)

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= cos^{1 + 1} (x)

Sumar:

1 + 1 = 2

= cos^{2} (x)

= cos^{2} (x) + sin^{2} (x)

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

cos^{2} (x) + sin^{2} (x) = 1

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

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

Re-escribir usando identidades trigonométricas

Utilizar la identidad trigonométrica básica:

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

\frac{1}{\frac{1}{s e c ( x )}}

Simplificar

\frac{1}{\frac{1}{s e c ( x )}}

Aplicar las propiedades de las fracciones:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

Aplicar la regla

\frac{a}{1} = a

= sec (x)

sec (x)

sec (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero