What is the general solution for (cot(x))/(csc(+1))=(csc(-1))/(cot(x)) ?

The general solution for (cot(x))/(csc(+1))=(csc(-1))/(cot(x)) is No Solution for x\in\mathbb{R}

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(cot(x))/(csc(+1))=(csc(-1))/(cot(x))

Solution

\frac{cot ( x )}{csc ( + 1 )} = \frac{csc ( - 1 )}{cot ( x )}

Solution

No Solution for x \in R

Solution steps

\frac{cot ( x )}{csc ( 1 )} = \frac{csc ( - 1 )}{cot ( x )}

csc (- 1) = - csc (1)

csc (- 1)

Use the following property:

csc (- x) = - csc (x)

csc (- 1) = - csc (1)

= - csc (1)

\frac{cot ( x )}{csc ( 1 )} = \frac{- csc ( 1 )}{cot ( x )}

Solve by substitution

\frac{cot ( x )}{csc ( 1 )} = \frac{- csc ( 1 )}{cot ( x )}

Let:

cot (x) = u

\frac{u}{csc ( 1 )} = \frac{- csc ( 1 )}{u}

\frac{u}{csc ( 1 )} = \frac{- csc ( 1 )}{u} : u = i csc (1), u = - i csc (1)

\frac{u}{csc ( 1 )} = \frac{- csc ( 1 )}{u}

Cross multiply

\frac{u}{csc ( 1 )} = \frac{- csc ( 1 )}{u}

Simplify

\frac{- csc ( 1 )}{u} : - \frac{csc ( 1 )}{u}

\frac{- csc ( 1 )}{u}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{csc ( 1 )}{u}

\frac{u}{csc ( 1 )} = - \frac{csc ( 1 )}{u}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

uu = - csc (1) csc (1)

Simplify

uu = - csc (1) csc (1)

Simplify

uu : u^{2}

uu

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

- csc (1) csc (1) : - csc^{2} (1)

- csc (1) csc (1)

Apply exponent rule:

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

csc (1) csc (1) = csc^{1 + 1} (1)

= - csc^{1 + 1} (1)

Add the numbers:

1 + 1 = 2

= - csc^{2} (1)

u^{2} = - csc^{2} (1)

u^{2} = - csc^{2} (1)

u^{2} = - csc^{2} (1)

Solve

u^{2} = - csc^{2} (1) : u = i csc (1), u = - i csc (1)

u^{2} = - csc^{2} (1)

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - csc^{2} (1), u = - - csc^{2} (1)

Simplify

- csc^{2} (1) : i csc (1)

- csc^{2} (1)

Apply radical rule:

- a = - 1 a

- csc^{2} (1) = - 1 csc^{2} (1)

= - 1 csc^{2} (1)

Apply imaginary number rule:

- 1 = i

= i csc^{2} (1)

Apply radical rule:

assuming

a \geq 0

csc^{2} (1) = csc (1)

= i csc (1)

Simplify

- - csc^{2} (1) : - i csc (1)

- - csc^{2} (1)

Simplify

- csc^{2} (1) : i csc (1)

- csc^{2} (1)

Apply radical rule:

- a = - 1 a

- csc^{2} (1) = - 1 csc^{2} (1)

= - 1 csc^{2} (1)

Apply imaginary number rule:

- 1 = i

= i csc^{2} (1)

Apply radical rule:

assuming

a \geq 0

csc^{2} (1) = csc (1)

= i csc (1)

= - i csc (1)

u = i csc (1), u = - i csc (1)

u = i csc (1), u = - i csc (1)

Substitute back

u = cot (x)

cot (x) = i csc (1), cot (x) = - i csc (1)

cot (x) = i csc (1), cot (x) = - i csc (1)

cot (x) = i csc (1) :

No Solution

cot (x) = i csc (1)

No Solution

cot (x) = - i csc (1) :

No Solution

cot (x) = - i csc (1)

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for (cot(x))/(csc(+1))=(csc(-1))/(cot(x)) ?
The general solution for (cot(x))/(csc(+1))=(csc(-1))/(cot(x)) is No Solution for x\in\mathbb{R}