What is the general solution for 4csc^2(x)-2=0 ?

The general solution for 4csc^2(x)-2=0 is No Solution for x\in\mathbb{R}

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4csc^2(x)-2=0

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

No Solution for x \in R

Solution steps

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

Solve by substitution

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

Let:

csc (x) = u

4 u^{2} - 2 = 0

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

4 u^{2} - 2 = 0

Move

2

to the right side

4 u^{2} - 2 = 0

Add

2

to both sides

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

Simplify

4 u^{2} = 2

4 u^{2} = 2

Divide both sides by

4

4 u^{2} = 2

Divide both sides by

4

\frac{4 u ^{2}}{4} = \frac{2}{4}

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

sin (2 θ - \frac{π}{2}) = - 1, 0 \leq θ < 2 π

18 cos (x) - 93 = 0

2 sin (3 θ) - 1 = 0

cot^{2} (θ) + 1 = \frac{3}{4}

- 2 cos (x) - cos (2 x) \cdot 2 = 0

What is the general solution for 4csc^2(x)-2=0 ?
The general solution for 4csc^2(x)-2=0 is No Solution for x\in\mathbb{R}