What is the general solution for (tan^2(3x))(csc(3x))+1=-1*sec^2(3x) ?

The general solution for (tan^2(3x))(csc(3x))+1=-1*sec^2(3x) is No Solution for x\in\mathbb{R}

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(tan^2(3x))(csc(3x))+1=-1*sec^2(3x)

(tan^{2} (3 x)) (csc (3 x)) + 1 = - 1 \cdot sec^{2} (3 x)

No Solution for x \in R

Solution steps

(tan^{2} (3 x)) (csc (3 x)) + 1 = - 1 \cdot sec^{2} (3 x)

Subtract

- 1 sec^{2} (3 x)

from both sides

tan^{2} (3 x) csc (3 x) + 1 + sec^{2} (3 x) = 0

Express with sin, cos

(\frac{sin ( 3 x )}{cos ( 3 x )})^{2} \frac{1}{sin ( 3 x )} + 1 + (\frac{1}{cos ( 3 x )})^{2} = 0

Simplify

(\frac{sin ( 3 x )}{cos ( 3 x )})^{2} \frac{1}{sin ( 3 x )} + 1 + (\frac{1}{cos ( 3 x )})^{2} : \frac{sin ( 3 x ) + 1 + cos ^{2} ( 3 x )}{cos ^{2} ( 3 x )}

\frac{sin ( 3 x ) + 1 + cos ^{2} ( 3 x )}{cos ^{2} ( 3 x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin (3 x) + 1 + cos^{2} (3 x) = 0

Subtract

cos^{2} (3 x)

from both sides

sin (3 x) + 1 = - cos^{2} (3 x)

Square both sides

(sin (3 x) + 1)^{2} = (- cos^{2} (3 x))^{2}

Subtract

(- cos^{2} (3 x))^{2}

from both sides

(sin (3 x) + 1)^{2} - cos^{4} (3 x) = 0

Factor

(sin (3 x) + 1)^{2} - cos^{4} (3 x) : (sin (3 x) + 1 + cos^{2} (3 x)) (sin (3 x) + 1 - cos^{2} (3 x))

(sin (3 x) + 1 + cos^{2} (3 x)) (sin (3 x) + 1 - cos^{2} (3 x)) = 0

Solving each part separately

sin (3 x) + 1 + cos^{2} (3 x) = 0 or sin (3 x) + 1 - cos^{2} (3 x) = 0

sin (3 x) + 1 + cos^{2} (3 x) = 0 : x = \frac{π}{2} + \frac{2 πn}{3}

sin (3 x) + 1 - cos^{2} (3 x) = 0 : x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

Combine all the solutions

x = \frac{π}{2} + \frac{2 πn}{3}, x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

Verify solutions by plugging them into the original equation

x = \frac{π}{2} + \frac{2 πn}{3}

Since the equation is undefined for:

\frac{π}{2} + \frac{2 πn}{3}

No Solution for x \in R

\frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} = 3

7 = 10 sin (40 (0 - h)) + 17

sin (2 x) = cos (x), 0 \leq x < 2 π

sec (x) = - \frac{5}{3}

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

What is the general solution for (tan^2(3x))(csc(3x))+1=-1*sec^2(3x) ?
The general solution for (tan^2(3x))(csc(3x))+1=-1*sec^2(3x) is No Solution for x\in\mathbb{R}