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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

sec (x) - sin (x) tan (x)

Expresar con seno, coseno

sec (x) - sin (x) tan (x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

\frac{1}{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 )}

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

Aplicar la regla

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Eliminar los terminos comunes:

cos (x)

= cos (x)

= cos (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

cos^{2} (8 x) - sin^{2} (8 x) = cos (16 x)

cos (2 x) cot (x) = 1 - cos^{2} (x)

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

- sin^{2} (θ) = cos^{2} (θ) - 1

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