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verificar sec^2(x)= 1/(1-sin^2(x))

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

sec^{2} (x)

Expresar con seno, coseno

sec^{2} (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

cos (\frac{π}{2} - x) sec (x) = tan (x)

cos (- x) + sin (- x) = cos (x) - sin (x)

\frac{1 - cos ( 2 θ )}{cos ( 2 θ ) + 1} = tan^{2} (θ)

cos (x - \frac{π}{3}) - sin (x + \frac{π}{6}) = 0

cos (x) - sin (x) = 2 \cdot cos (x + \frac{π}{4})