What is the general solution for cos^2(x)+1=4 ?

The general solution for cos^2(x)+1=4 is No Solution for x\in\mathbb{R}

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cos^2(x)+1=4

cos^{2} (x) + 1 = 4

No Solution for x \in R

Solution steps

cos^{2} (x) + 1 = 4

Solve by substitution

cos^{2} (x) + 1 = 4

Let:

cos (x) = u

u^{2} + 1 = 4

u^{2} + 1 = 4 : u = 3, u = - 3

u^{2} + 1 = 4

Move

1

to the right side

u^{2} + 1 = 4

Subtract

1

from both sides

u^{2} + 1 - 1 = 4 - 1

Simplify

u^{2} = 3

u^{2} = 3

For

x^{2} = f (a)

the solutions are

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

u = 3, u = - 3

Substitute back

u = cos (x)

cos (x) = 3, cos (x) = - 3

cos (x) = 3, cos (x) = - 3

cos (x) = 3 :

No Solution

cos (x) = 3

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = - 3 :

No Solution

cos (x) = - 3

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

1 = sin^{2} (x)

- 2 sin^{2} (x) - cos (x) = - 1

tan^{2} (x) = 1 + sec (x)

8 sin^{2} (x) = 7

- 6 arcsin (3 x) = π

What is the general solution for cos^2(x)+1=4 ?
The general solution for cos^2(x)+1=4 is No Solution for x\in\mathbb{R}