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

The general solution for cot^2(x)=tan(x/2) is x=2*0.47448…+2pin,x=2*1.34922…+2pin

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cot^2(x)=tan(x/2)

cot^{2} (x) = tan (\frac{x}{2})

x = 2 \cdot 0.47448 \dots + 2 πn, x = 2 \cdot 1.34922 \dots + 2 πn

Solution steps

cot^{2} (x) = tan (\frac{x}{2})

Subtract

tan (\frac{x}{2})

from both sides

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

Let:

u = \frac{x}{2}

cot^{2} (2 u) - tan (u) = 0

Rewrite using trig identities

\frac{( 1 - tan ^{2} ( u ) ) ^{2}}{4 tan ^{2} ( u )} - tan (u) = 0

Solve by substitution

tan (u) \approx 0.51361 \dots, tan (u) \approx 4.43910 \dots

tan (u) = 0.51361 \dots : u = arctan (0.51361 \dots) + πn

tan (u) = 4.43910 \dots : u = arctan (4.43910 \dots) + πn

Combine all the solutions

u = arctan (0.51361 \dots) + πn, u = arctan (4.43910 \dots) + πn

Substitute back

u = \frac{x}{2}

\frac{x}{2} = arctan (0.51361 \dots) + πn : x = 2 arctan (0.51361 \dots) + 2 πn

\frac{x}{2} = arctan (4.43910 \dots) + πn : x = 2 arctan (4.43910 \dots) + 2 πn

x = 2 arctan (0.51361 \dots) + 2 πn, x = 2 arctan (4.43910 \dots) + 2 πn

Show solutions in decimal form

x = 2 \cdot 0.47448 \dots + 2 πn, x = 2 \cdot 1.34922 \dots + 2 πn

cos (x) = \frac{15}{21}

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

2 cos (x) - 1 = 0, 0 \leq x < 2 π

cos (t) = \frac{3}{5}

cos (4 θ) = - \frac{2}{3}

What is the general solution for cot^2(x)=tan(x/2) ?
The general solution for cot^2(x)=tan(x/2) is x=2*0.47448…+2pin,x=2*1.34922…+2pin