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verificar sin^2(2x)= 1/2-1/2 cos(4x)

Solución

verificar

sin^{2} (2 x) = \frac{1}{2} - \frac{1}{2} cos (4 x)

Verdadero

Pasos de solución

sin^{2} (2 x) = \frac{1}{2} - \frac{1}{2} cos (4 x)

Manipular el lado izquierdo

\frac{1}{2} - \frac{1}{2} cos (4 x)

Re-escribir usando identidades trigonométricas

\frac{1}{2} - \frac{1}{2} cos (4 x)

= \frac{1}{2} - \frac{1}{2} cos (2 \cdot 2 x)

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

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

Expandir

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

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

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = - \frac{1}{2}, b = 1, c = 2 sin^{2} (2 x)

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

Aplicar las reglas de los signos

- (- a) = a

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

Simplificar

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

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

1 \cdot \frac{1}{2} = \frac{1}{2}

1 \cdot \frac{1}{2}

Multiplicar:

1 \cdot \frac{1}{2} = \frac{1}{2}

= \frac{1}{2}

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

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

Multiplicar:

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

= sin^{2} (2 x)

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

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

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

Sumar elementos similares:

\frac{1}{2} - \frac{1}{2} = 0

= sin^{2} (2 x)

= sin^{2} (2 x)

= sin^{2} (2 x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

sin (\frac{a}{2}) = \frac{1}{2} sin (a)

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

tan (x) sec (x) (csc (x) - sin (x)) = 1

csc (x) cos (x) cot (x) = sin (x)

sin^{4} (x) + cos^{4} (x) = 1