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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado izquierdo

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

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

Simplificar

sin^{2} (x) + sin^{2} (x) (sec^{2} (x) - 1) : sin^{2} (x) sec^{2} (x)

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

Expandir

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

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

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = sin^{2} (x), b = sec^{2} (x), c = 1

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

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

Multiplicar:

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

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

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

Sumar elementos similares:

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

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

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

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

Expresar con seno, coseno

sec^{2} (x) sin^{2} (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

(\frac{1}{cos ( x )})^{2} sin^{2} (x) : \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

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

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

(\frac{1}{cos ( x )})^{2}

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Multiplicar:

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica básica:

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

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

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

Utilizar la identidad trigonométrica básica:

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

tan (x) tan (x)

Simplificar

tan (x) tan (x) : tan^{2} (x)

tan (x) tan (x)

Aplicar las leyes de los exponentes:

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

tan (x) tan (x) = tan^{1 + 1} (x)

= tan^{1 + 1} (x)

Sumar:

1 + 1 = 2

= tan^{2} (x)

tan^{2} (x)

tan^{2} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

- sin (x) sec (x) = - tan (x)

cos (4 x) - cos (3 x) cos (x) = sin (2 x) - 4 cos (2 x) sin (2 x)

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

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

\frac{( 1 )}{( 2 )} (cot (θ) - tan (θ)) = cot (2 θ)