What is the general solution for csc^2(x)+2cos(x)+1=0 ?

The general solution for csc^2(x)+2cos(x)+1=0 is No Solution for x\in\mathbb{R}

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csc^2(x)+2cos(x)+1=0

csc^{2} (x) + 2 cos (x) + 1 = 0

No Solution for x \in R

Solution steps

csc^{2} (x) + 2 cos (x) + 1 = 0

Subtract

2 cos (x)

from both sides

csc^{2} (x) + 1 = - 2 cos (x)

Square both sides

(csc^{2} (x) + 1)^{2} = (- 2 cos (x))^{2}

Subtract

(- 2 cos (x))^{2}

from both sides

(csc^{2} (x) + 1)^{2} - 4 cos^{2} (x) = 0

Rewrite using trig identities

(1 + \frac{1}{sin ^{2} ( x )})^{2} - 4 cos^{2} (x) = 0

Factor

(1 + \frac{1}{sin ^{2} ( x )})^{2} - 4 cos^{2} (x) : (1 + \frac{1}{sin ^{2} ( x )} + 2 cos (x)) (1 + \frac{1}{sin ^{2} ( x )} - 2 cos (x))

(1 + \frac{1}{sin ^{2} ( x )} + 2 cos (x)) (1 + \frac{1}{sin ^{2} ( x )} - 2 cos (x)) = 0

Solving each part separately

1 + \frac{1}{sin ^{2} ( x )} + 2 cos (x) = 0 or 1 + \frac{1}{sin ^{2} ( x )} - 2 cos (x) = 0

1 + \frac{1}{sin ^{2} ( x )} + 2 cos (x) = 0 :

No Solution

1 + \frac{1}{sin ^{2} ( x )} - 2 cos (x) = 0 :

No Solution

Combine all the solutions

No Solution

Verify solutions by plugging them into the original equation

No Solution for x \in R

2 cos^{2} (2 x) + 3 cos (2 x) - 2 = 0

tan (t) = - \frac{12}{5}, \frac{3 π}{2} < t < 2 π

3 sin (x) + 5 cos (x) = 4

- 2 = sec (x)

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

What is the general solution for csc^2(x)+2cos(x)+1=0 ?
The general solution for csc^2(x)+2cos(x)+1=0 is No Solution for x\in\mathbb{R}