What is the general solution for 2csc^2(x)-1=0 ?

The general solution for 2csc^2(x)-1=0 is No Solution for x\in\mathbb{R}

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2csc^2(x)-1=0

2 csc^{2} (x) - 1 = 0

No Solution for x \in R

Solution steps

2 csc^{2} (x) - 1 = 0

Solve by substitution

2 csc^{2} (x) - 1 = 0

Let:

csc (x) = u

2 u^{2} - 1 = 0

2 u^{2} - 1 = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

2 u^{2} - 1 = 0

Move

1

to the right side

2 u^{2} - 1 = 0

Add

1

to both sides

2 u^{2} - 1 + 1 = 0 + 1

Simplify

2 u^{2} = 1

2 u^{2} = 1

Divide both sides by

2

2 u^{2} = 1

Divide both sides by

2

\frac{2 u ^{2}}{2} = \frac{1}{2}

Simplify

u^{2} = \frac{1}{2}

u^{2} = \frac{1}{2}

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{2}, u = - \frac{1}{2}

Substitute back

u = csc (x)

csc (x) = \frac{1}{2}, csc (x) = - \frac{1}{2}

csc (x) = \frac{1}{2}, csc (x) = - \frac{1}{2}

csc (x) = \frac{1}{2} :

No Solution

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

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

csc (x) = - \frac{1}{2} :

No Solution

csc (x) = - \frac{1}{2}

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

Combine all the solutions

No Solution for x \in R

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What is the general solution for 2csc^2(x)-1=0 ?
The general solution for 2csc^2(x)-1=0 is No Solution for x\in\mathbb{R}