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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

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

Aplicar la formula del binomio al cuadrado:

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

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

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Aplicar las leyes de los exponentes:

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

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

Multiplicar los numeros:

2 \cdot 2 = 4

= cos^{4} (x)

= cos^{4} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

1 + sec (x) = \frac{cos ( x ) + 1}{cos ( x )}

tan (θ) + cot (θ) = csc (θ) sec (θ)

2 sin (x) + 1 = 3 csc (x)

sec (x) cot (x) = \frac{1}{sin ( x )}

sec (2 θ) = \frac{sec ^{2} ( θ ) csc ^{2} ( θ )}{csc ^{2} ( θ ) - sec ^{2} ( θ )}