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verificar 1+1/(tan^2(a))= 1/(sin^2(a))

Solución

verificar

1 + \frac{1}{tan ^{2} ( a )} = \frac{1}{sin ^{2} ( a )}

Verdadero

Pasos de solución

1 + \frac{1}{tan ^{2} ( a )} = \frac{1}{sin ^{2} ( a )}

Manipular el lado derecho

1 + \frac{1}{tan ^{2} ( a )}

Expresar con seno, coseno

1 + \frac{1}{tan ^{2} ( a )}

Utilizar la identidad trigonométrica básica:

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

= 1 + \frac{1}{( \frac{s i n ( a )}{c o s ( a )} ) ^{2}}

Simplificar

1 + \frac{1}{( \frac{s i n ( a )}{c o s ( a )} ) ^{2}} : \frac{sin ^{2} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

1 + \frac{1}{( \frac{s i n ( a )}{c o s ( a )} ) ^{2}}

\frac{1}{( \frac{s i n ( a )}{c o s ( a )} ) ^{2}} = \frac{cos ^{2} ( a )}{sin ^{2} ( a )}

\frac{1}{( \frac{s i n ( a )}{c o s ( a )} ) ^{2}}

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

(\frac{sin ( a )}{cos ( a )})^{2}

Aplicar las leyes de los exponentes:

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

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

= \frac{1}{\frac{s i n ^{2} ( a )}{c o s ^{2} ( a )}}

Aplicar las propiedades de las fracciones:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

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

Convertir a fracción:

1 = \frac{1 sin ^{2} ( a )}{sin ^{2} ( a )}

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

Multiplicar:

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

= \frac{1}{sin ^{2} ( a )}

= \frac{1}{sin ^{2} ( a )}

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

\frac{tan ( θ )}{cot ( θ )} = tan^{2} (θ)

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

\frac{sin ( 6 x )}{cos ( x )} = 6 sin (x)

csc (x) sec (x) - tan (x) = cot (x)

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