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(1+cos^2(x))/(sin^2(x))=1

Solución

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

x = \frac{π}{2} + πn

Grados

x = 9 0^{\circ} + 18 0^{\circ} n

Pasos de solución

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

Restar

1

de ambos lados

\frac{1 + cos ^{2} ( x )}{sin ^{2} ( x )} - 1 = 0

Simplificar

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

Multiplicar:

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

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

\frac{1 + cos ^{2} ( x ) - sin ^{2} ( x )}{sin ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

1 + cos^{2} (x) - sin^{2} (x) = 0

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

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

= 1 + cos (2 x)

1 + cos (2 x) = 0

Mover

1

al lado derecho

1 + cos (2 x) = 0

Restar

1

de ambos lados

1 + cos (2 x) - 1 = 0 - 1

Simplificar

cos (2 x) = - 1

cos (2 x) = - 1

Soluciones generales para

cos (2 x) = - 1

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

2 x = π + 2 πn

2 x = π + 2 πn

Resolver

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

Dividir ambos lados entre

2

2 x = π + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

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

w, 100 cos (10 w) = 50

cos (θ) = \frac{8}{25}

sin (2 x) = π

sin (x + 5 π) = 0