What is the general solution for sec^2(x)= 1/2 ?

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

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sec^2(x)= 1/2

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

No Solution for x \in R

Solution steps

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

Solve by substitution

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

Let:

sec (x) = u

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

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

3 sin^{2} (x) = 3 cos^{2} (x)

9 tan (\frac{x}{2}) + 7 = 5 tan (\frac{x}{2}) + 3

3 - 3 cos (θ) = 3

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

5 sin (θ) - 5 cos (θ) = 2

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