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verificar (sin(2x))/(cos(2x)+sin^2(x))=2tan(x)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

sin (2 x) = 2 sin (x) cos (x)

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

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

Sumar elementos similares:

- sin^{2} (x) + sin^{2} (x) = 0

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

Eliminar los terminos comunes:

cos (x)

= \frac{2 sin ( x )}{cos ( x )}

= \frac{2 sin ( x )}{cos ( x )}

= \frac{2 sin ( x )}{cos ( x )}

Re-escribir usando identidades trigonométricas

= 2 \cdot \frac{sin ( x )}{cos ( x )}

Utilizar la identidad trigonométrica básica:

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

2 tan (x)

2 tan (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

tan (θ) sec (θ) (csc (θ) - sin (θ)) = 1

7 cos^{2} (x) + 5 sin^{2} (x) = 6 + cos (2 x)

(1 - sin^{2} (A)) (1 + tan^{2} (A)) = 1

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

sin^{4} (θ) + cos^{2} (θ) sin^{2} (θ) = sin^{2} (θ)