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verificar 2cos(θ)-(cos(2θ))/(cos(θ))=sec(θ)

Solución

verificar

2 cos (θ) - \frac{cos ( 2 θ )}{cos ( θ )} = sec (θ)

Verdadero

Pasos de solución

2 cos (θ) - \frac{cos ( 2 θ )}{cos ( θ )} = sec (θ)

Manipular el lado izquierdo

sec (θ)

Expresar con seno, coseno

sec (θ)

Utilizar la identidad trigonométrica básica:

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

= \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )}

Re-escribir usando identidades trigonométricas

\frac{1}{cos ( θ )}

Utilizar la identidad trigonométrica del ángulo doble:

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

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

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

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

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

Expandir

\frac{- cos ( 2 θ ) + 2 cos ^{2} ( θ )}{cos ( θ )} : - \frac{cos ( 2 θ )}{cos ( θ )} + 2 cos (θ)

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

Aplicar las propiedades de las fracciones:

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

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

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

Cancelar

\frac{2 cos ^{2} ( θ )}{cos ( θ )} : 2 cos (θ)

\frac{2 cos ^{2} ( θ )}{cos ( θ )}

Eliminar los terminos comunes:

cos (θ)

= 2 cos (θ)

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

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

= 2 cos (θ) - \frac{cos ( 2 θ )}{cos ( θ )}

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

cos (2 \cdot 30) = cos^{2} (30) - sin^{2} (30)

1 + tan^{2} (4 5^{\circ}) = sec^{2} (4 5^{\circ})

\frac{cot ( x )}{cos ( x )} = csc (x)

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

2 sin^{2} (2 t) + cos (4 t) = 1