What is the general solution for cot^2(x)= 3/2 csc(x) ?

The general solution for cot^2(x)= 3/2 csc(x) is x= pi/6+2pin,x=(5pi)/6+2pin

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cot^2(x)= 3/2 csc(x)

cot^{2} (x) = \frac{3}{2} csc (x)

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

cot^{2} (x) = \frac{3}{2} csc (x)

Subtract

\frac{3}{2} csc (x)

from both sides

cot^{2} (x) - \frac{3}{2} csc (x) = 0

Simplify

cot^{2} (x) - \frac{3}{2} csc (x) : \frac{2 cot ^{2} ( x ) - 3 csc ( x )}{2}

\frac{2 cot ^{2} ( x ) - 3 csc ( x )}{2} = 0

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

2 cot^{2} (x) - 3 csc (x) = 0

Rewrite using trig identities

(- 1 + csc^{2} (x)) \cdot 2 - 3 csc (x) = 0

Solve by substitution

csc (x) = 2, csc (x) = - \frac{1}{2}

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = - \frac{1}{2} :

No Solution

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 cos (θ) - 1 = 0, 0 \leq θ \leq 2 π

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

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

sin (\frac{π x}{4}) = 0

tan^{2} (x) = - 1 - 2 tan (- x)

What is the general solution for cot^2(x)= 3/2 csc(x) ?
The general solution for cot^2(x)= 3/2 csc(x) is x= pi/6+2pin,x=(5pi)/6+2pin