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verificar cos^2(2x)= 1/2+1/2 cos(4x)

Solución

verificar

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

Verdadero

Pasos de solución

cos^{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)

Simplificar

\frac{1}{2} + cos (4 x) \frac{1}{2} : \frac{1 + cos ( 4 x )}{2}

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

Multiplicar

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

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

Multiplicar fracciones:

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

= \frac{cos ( 4 x )}{2}

= \frac{1}{2} + \frac{cos ( 4 x )}{2}

Aplicar la regla

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

= \frac{1 + cos ( 4 x )}{2}

= \frac{1 + cos ( 4 x )}{2}

Re-escribir usando identidades trigonométricas

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

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

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

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

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

Agrupar términos semejantes

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

1 - 1 = 0

= 2 cos^{2} (2 x)

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

Dividir:

\frac{2}{2} = 1

= cos^{2} (2 x)

= cos^{2} (2 x)

= cos^{2} (2 x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

\frac{csc ( x ) - 1}{1 - sin ( x )} = csc (x)

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

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

2 sin (- t) cos (- t) = 2 sin (- t) cos (t)