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verificar (cos(x)-sin(x))^2=1-2sin(x)*cos(x)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

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

Expandir

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

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

Aplicar la formula del binomio al cuadrado:

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

a = cos (x), b = sin (x)

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

= 1 - 2 cos (x) sin (x)

= 1 - 2 cos (x) sin (x)

= 1 - 2 sin (x) cos (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

tan^{3} (x) = \frac{sin ^{3} ( x )}{cos ^{3} ( x )}

cos (4 θ) = 1 - 8 sin^{2} (θ) cos^{2} (θ)

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

cos^{2} (θ) = sin^{2} (θ) - 1