What is the general solution for sin^2(θ)=-1 ?

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

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sin^2(θ)=-1

sin^{2} (θ) = - 1

No Solution for θ \in R

Solution steps

sin^{2} (θ) = - 1

Solve by substitution

sin^{2} (θ) = - 1

Let:

sin (θ) = u

u^{2} = - 1

u^{2} = - 1 : u = i, u = - i

u^{2} = - 1

For

x^{2} = f (a)

the solutions are

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

u = - 1, u = - - 1

Simplify

- 1 : i

- 1

Apply imaginary number rule:

- 1 = i

= i

Simplify

- - 1 : - i

- - 1

Apply imaginary number rule:

- 1 = i

= - i

u = i, u = - i

Substitute back

u = sin (θ)

sin (θ) = i, sin (θ) = - i

sin (θ) = i, sin (θ) = - i

sin (θ) = i :

No Solution

sin (θ) = i

No Solution

sin (θ) = - i :

No Solution

sin (θ) = - i

No Solution

Combine all the solutions

No Solution for θ \in R

10 sec (2 x) + 5 tan (2 x) - 15 = 0

- tan (x) + sec (x) = 1, 0 \leq x \leq 2 π

3 tan (\frac{x}{9}) - 3 = 0

so l v e f or x, tan (x) = \frac{1}{3}

cos (x) = \frac{9}{12}

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