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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado izquierdo

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

arccsc (x) = sin (x)

cot (θ) = \frac{sin ( θ )}{cos ( θ )}

(1 - tan^{2} (θ))^{2} = sec^{2} (θ) - 4 tan^{2} (θ)

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

4 sin (2 x) = 8 sin (x) cos (x)