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verificar (tan(θ))/(cot(θ))=tan^2(θ)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

\frac{tan ( θ )}{cot ( θ )}

Expresar con seno, coseno

\frac{tan ( θ )}{cot ( θ )}

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{s i n ( θ )}{c o s ( θ )}}{cot ( θ )}

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{s i n ( θ )}{c o s ( θ )}}{\frac{c o s ( θ )}{s i n ( θ )}}

Simplificar

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

\frac{\frac{s i n ( θ )}{c o s ( θ )}}{\frac{c o s ( θ )}{s i n ( θ )}}

Dividir fracciones:

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

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

sin (θ) sin (θ) = sin^{2} (θ)

sin (θ) sin (θ)

Aplicar las leyes de los exponentes:

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

sin (θ) sin (θ) = sin^{1 + 1} (θ)

= sin^{1 + 1} (θ)

Sumar:

1 + 1 = 2

= sin^{2} (θ)

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

cos (θ) cos (θ) = cos^{2} (θ)

cos (θ) cos (θ)

Aplicar las leyes de los exponentes:

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

cos (θ) cos (θ) = cos^{1 + 1} (θ)

= cos^{1 + 1} (θ)

Sumar:

1 + 1 = 2

= cos^{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

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 )}

\frac{1}{sec ( x ) ( 1 - sin ( x ) )} = sec (x) + tan (x)