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

Solución

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

Solución

Sin soluci \overset{o}{ˊ} n para x \in R

Pasos de solución

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

Restar

sec (x) csc (x)

de ambos lados

tan (x) cot (x) - sec (x) csc (x) = 0

Expresar con seno, coseno

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

Utilizar la identidad trigonométrica básica:

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

= \frac{cos ( x )}{sin ( x )} tan (x) - csc (x) sec (x)

Utilizar la identidad trigonométrica básica:

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

= \frac{cos ( x )}{sin ( x )} \cdot \frac{sin ( x )}{cos ( x )} - csc (x) sec (x)

Utilizar la identidad trigonométrica básica:

csc (x) = \frac{1}{sin ( x )}

= \frac{cos ( x )}{sin ( x )} \cdot \frac{sin ( x )}{cos ( x )} - \frac{1}{sin ( x )} sec (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

\frac{cos ( x )}{sin ( x )} \cdot \frac{sin ( x )}{cos ( x )} - \frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )} : \frac{sin ( x ) cos ( x ) - 1}{sin ( x ) cos ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{sin ( x )}{cos ( x )} - \frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{sin ( x )}{cos ( x )} = 1

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos (x)

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

Eliminar los terminos comunes:

sin (x)

= 1

\frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )} = \frac{1}{sin ( x ) cos ( x )}

\frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )}

Multiplicar fracciones:

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

= \frac{1 \cdot 1}{sin ( x ) cos ( x )}

Multiplicar los numeros:

1 \cdot 1 = 1

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

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

Convertir a fracción:

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

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

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 ( x ) cos ( x ) - 1}{sin ( x ) cos ( x )}

Multiplicar:

1 \cdot sin (x) = sin (x)

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

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

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

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

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

Re-escribir usando identidades trigonométricas

- 1 + cos (x) sin (x)

Utilizar la identidad trigonométrica del ángulo doble:

2 sin (x) cos (x) = sin (2 x)

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

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

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

Mover

1

al lado derecho

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

Sumar

1

a ambos lados

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

Simplificar

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

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

Multiplicar ambos lados por

2

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

Multiplicar ambos lados por

2

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

Simplificar

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

Sin soluci \overset{o}{ˊ} n para x \in R

Ejemplos populares

4sin(x)=cos(x)-2

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

sin(θ)=1-cos(θ)

sin (θ) = 1 - cos (θ)

cos(θ)-2sin(θ)cos(θ)=0

cos (θ) - 2 sin (θ) cos (θ) = 0

solvefor x,cos(xy)=sin(x+y)resolver para

x, cos (x y) = sin (x + y)

cot(x)= 3/4

cot (x) = \frac{3}{4}