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

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

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4sec^2(x)-2=0

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

No Solution for x \in R

Solution steps

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

Solve by substitution

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

Let:

sec (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 = sec (x)

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

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

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

No Solution

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

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

No Solution

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

No Solution

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

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

No Solution

Combine all the solutions

No Solution for x \in R

cos (x) = \frac{1}{3}

sin (x) cos (x) = \frac{3}{4}, 0 \leq x \leq 2 π

2 cos^{2} (x) + sin (x) - 1 = 0, 0 \leq x < 2 π

cot (x) = - 1, 0 \leq x \leq 2 π

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

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