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verificar sin^5(x)=(1-2cos^2(x)+cos^4(x))(sin(x))

Solución

verificar

sin^{5} (x) = (1 - 2 cos^{2} (x) + cos^{4} (x)) (sin (x))

Verdadero

Pasos de solución

sin^{5} (x) = (1 - 2 cos^{2} (x) + cos^{4} (x)) sin (x)

Manipular el lado izquierdo

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

Re-escribir usando identidades trigonométricas

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

Aplicar la formula del binomio al cuadrado:

a^{2} - 2 ab + b^{2} = (a - b)^{2}

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

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Aplicar las leyes de los exponentes:

(a^{b})^{c} = a^{b c}

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

Multiplicar los numeros:

2 \cdot 2 = 4

= sin^{4} (x)

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

Simplificar

sin^{4} (x) sin (x) : sin^{5} (x)

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

Aplicar las leyes de los exponentes:

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

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

= sin^{4 + 1} (x)

Sumar:

4 + 1 = 5

= sin^{5} (x)

= sin^{5} (x)

= sin^{5} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

2 cos^{2} (x) = 1

\frac{cos ( a )}{sin ( a ) cot ( a )} = 1

2 (cos (3 θ) + 3 cos (θ)) = 8 cos^{3} (θ)

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