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verificar 1/(cos^2(x))=1+tan^2(x)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado izquierdo

1 + tan^{2} (x)

Expresar con seno, coseno

1 + tan^{2} (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

Multiplicar:

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

sin (θ - \frac{3 π}{2}) = cos (θ)

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

sin (27 0^{\circ} + x) = - cos (x)

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