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verificar sec(x)(sec(x)-cos(x))=tan^2(x)

Solución

verificar

sec (x) (sec (x) - cos (x)) = tan^{2} (x)

Verdadero

Pasos de solución

sec (x) (sec (x) - cos (x)) = tan^{2} (x)

Manipular el lado derecho

sec (x) (sec (x) - cos (x))

Expandir

(- cos (x) + sec (x)) sec (x) : - sec (x) cos (x) + sec^{2} (x)

(- cos (x) + sec (x)) sec (x)

= sec (x) (- cos (x) + sec (x))

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = sec (x), b = - cos (x), c = sec (x)

= sec (x) (- cos (x)) + sec (x) sec (x)

Aplicar las reglas de los signos

+ (- a) = - a

= - sec (x) cos (x) + sec (x) sec (x)

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

sec (x) sec (x)

Aplicar las leyes de los exponentes:

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

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

= sec^{1 + 1} (x)

Sumar:

1 + 1 = 2

= sec^{2} (x)

= - sec (x) cos (x) + sec^{2} (x)

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

Re-escribir usando identidades trigonométricas

sec^{2} (x) - cos (x) sec (x)

sec (x) cos (x) = 1

sec (x) cos (x)

Expresar con seno, coseno

sec (x) cos (x)

Utilizar la identidad trigonométrica básica:

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

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

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos (x)

= 1

= 1

= sec^{2} (x) - 1

Utilizar la identidad pitagórica:

sec^{2} (x) = tan^{2} (x) + 1

sec^{2} (x) - 1 = tan^{2} (x)

= tan^{2} (x)

= tan^{2} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

sin (3 x) = 3 sin (x) - 4 (sin (x))^{3}

(5 sin (θ) - 5 cos (θ))^{2} = 25 - 25 sin (2 θ)

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

cot (x) \cdot sec (x) = csc (x)