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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado izquierdo

\frac{1 - cos ( 2 x )}{1 + cos ( 2 x )}

Re-escribir usando identidades trigonométricas

\frac{1 - cos ( 2 x )}{1 + cos ( 2 x )}

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

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

Sumar:

1 + 1 = 2

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

Expandir

1 - (1 - 2 sin^{2} (x)) : 2 sin^{2} (x)

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

- (1 - 2 sin^{2} (x)) : - 1 + 2 sin^{2} (x)

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

Poner los parentesis

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

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

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

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

1 - 1 = 0

= 2 sin^{2} (x)

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

Factorizar

- 2 sin^{2} (x) + 2 : 2 (- sin^{2} (x) + 1)

- 2 sin^{2} (x) + 2

Reescribir como

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

Factorizar el termino común

2

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

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

Dividir:

\frac{2}{2} = 1

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

Quitar los parentesis:

(- a) = - a

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

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

Utilizar la identidad pitagórica:

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica básica:

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

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

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

Utilizar la identidad trigonométrica básica:

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

tan (x) tan (x)

Simplificar

tan (x) tan (x) : tan^{2} (x)

tan (x) tan (x)

Aplicar las leyes de los exponentes:

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

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

= tan^{1 + 1} (x)

Sumar:

1 + 1 = 2

= tan^{2} (x)

tan^{2} (x)

tan^{2} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

sin (2 x) = 2 sin (x)

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

sin (θ) - cos^{2} (θ) = sin^{3} (θ)

1 - tan^{2} (a) = cos (2 a) sec^{2} (a)

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