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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

sin^{2} (x) - (1 - sin^{2} (x)) sin^{2} (x) : sin^{4} (x)

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

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

Expandir

- sin^{2} (x) (1 - sin^{2} (x)) : - sin^{2} (x) + sin^{4} (x)

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

Poner los parentesis utilizando:

a (b - c) = ab - a c

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

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

Aplicar las reglas de los signos

- (- a) = a

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

Simplificar

- 1 \cdot sin^{2} (x) + sin^{2} (x) sin^{2} (x) : - sin^{2} (x) + sin^{4} (x)

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

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

1 \cdot sin^{2} (x)

Multiplicar:

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

= sin^{2} (x)

sin^{2} (x) sin^{2} (x) = sin^{4} (x)

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

Aplicar las leyes de los exponentes:

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

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

= sin^{2 + 2} (x)

Sumar:

2 + 2 = 4

= sin^{4} (x)

= - sin^{2} (x) + sin^{4} (x)

= - sin^{2} (x) + sin^{4} (x)

= sin^{2} (x) - sin^{2} (x) + sin^{4} (x)

Sumar elementos similares:

sin^{2} (x) - sin^{2} (x) = 0

= sin^{4} (x)

= sin^{4} (x)

= sin^{4} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

sin (2 θ) = \frac{1}{2}

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

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

- tan^{2} (x) = 1 - sec^{2} (x)

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