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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

sin^{2} (θ) + sin^{2} (θ) tan^{2} (θ)

Factorizar

sin^{2} (θ) + sin^{2} (θ) tan^{2} (θ) : sin^{2} (θ) (1 + tan^{2} (θ))

sin^{2} (θ) + sin^{2} (θ) tan^{2} (θ)

Factorizar el termino común

sin^{2} (θ)

= sin^{2} (θ) (1 + tan^{2} (θ))

= (1 + tan^{2} (θ)) sin^{2} (θ)

Re-escribir usando identidades trigonométricas

(1 + tan^{2} (θ)) sin^{2} (θ)

Utilizar la identidad pitagórica:

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

= sec^{2} (θ) sin^{2} (θ)

= sec^{2} (θ) sin^{2} (θ)

Expresar con seno, coseno

sec^{2} (θ) sin^{2} (θ)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

(\frac{1}{cos ( θ )})^{2} sin^{2} (θ)

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

(\frac{1}{cos ( θ )})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{1 ^{2}}{cos ^{2} ( θ )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

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

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot sin^{2} (θ) = sin^{2} (θ)

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica básica:

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

\frac{sin ( θ ) tan ( θ )}{cos ( θ )}

= tan (θ) \frac{sin ( θ )}{cos ( θ )}

Utilizar la identidad trigonométrica básica:

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

tan (θ) tan (θ)

Simplificar

tan (θ) tan (θ) : tan^{2} (θ)

tan (θ) tan (θ)

Aplicar las leyes de los exponentes:

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

tan (θ) tan (θ) = tan^{1 + 1} (θ)

= tan^{1 + 1} (θ)

Sumar:

1 + 1 = 2

= tan^{2} (θ)

tan^{2} (θ)

tan^{2} (θ)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

1 + sin (2 θ) = (cos (θ) sin (θ))^{2}

sec (θ) csc (θ) = tan (θ) + cot (θ)

2 tan (0^{\circ}) = sin (0^{\circ})

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

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