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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 izquierdo

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

Factorizar

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

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

Aplicar las leyes de los exponentes:

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

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

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

Factorizar el termino común

sin^{2} (x)

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

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

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

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

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