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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

Simplificar

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

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

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

Aplicar la formula del binomio al cuadrado:

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

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

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

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

Simplificar

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

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

Sumar elementos similares:

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

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

Sumar elementos similares:

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

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

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

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

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

= sin (2 x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

sin (t) cos (t) = 1

\frac{1}{1 - sin ( B )} = sec^{2} (B) + sec (B) tan (B)

tan (6 0^{\circ}) = \frac{2 tan ( 3 0 ^{\circ} )}{1 - tan ^{2} ( 3 0 ^{\circ} )}

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

cos (x) + sin (x) = 1