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verificar cos(x)cot(x)=(1-sin^2(x))/(sin(x))

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

cos (x) cot (x)

Expresar con seno, coseno

cos (x) cot (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Multiplicar fracciones:

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

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

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

cos (x) cos (x)

Aplicar las leyes de los exponentes:

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

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

= cos^{1 + 1} (x)

Sumar:

1 + 1 = 2

= cos^{2} (x)

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

2 cot (4 x) = cot (2 x) - tan (2 x)

sin (2 x) - cot (x) = - cot (x) cos (2 x)

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

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