What is the general solution for cot^2(x)=tan(1/2 x) ?

The general solution for cot^2(x)=tan(1/2 x) is x=2*0.47448…+2pin,x=2*1.34922…+2pin

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cot^2(x)=tan(1/2 x)

cot^{2} (x) = tan (\frac{1}{2} x)

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

Solution steps

cot^{2} (x) = tan (\frac{1}{2} x)

Subtract

tan (\frac{1}{2} x)

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 (C) = \frac{3}{4}

cos (x - \frac{π}{6}) - sin (x) = 0

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

6 sec^{2} (x) + tan (x) - 7 = 0

4 \cdot sin^{2} (x) = 3

What is the general solution for cot^2(x)=tan(1/2 x) ?
The general solution for cot^2(x)=tan(1/2 x) is x=2*0.47448…+2pin,x=2*1.34922…+2pin