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

The general solution for cot(x)(1-cos^2(x))= 1/2 sin^2(x) is x=1.10714…+pin

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cot(x)(1-cos^2(x))= 1/2 sin^2(x)

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

x = 1.10714 \dots + πn

Solution steps

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

Subtract

\frac{1}{2} sin^{2} (x)

from both sides

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

Simplify

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

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

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

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

Rewrite using trig identities

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

Factor

- sin^{2} (x) + 2 cot (x) sin^{2} (x) : sin^{2} (x) (2 cot (x) - 1)

sin^{2} (x) (2 cot (x) - 1) = 0

Solving each part separately

sin^{2} (x) = 0 or 2 cot (x) - 1 = 0

sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

2 cot (x) - 1 = 0 : x = arccot (\frac{1}{2}) + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccot (\frac{1}{2}) + πn

Since the equation is undefined for:

2 πn, π + 2 πn

x = arccot (\frac{1}{2}) + πn

Show solutions in decimal form

x = 1.10714 \dots + πn

(tan (x) - 1) (2 sin (x) - 1) = 0

cos (x) = \frac{20}{29}

2 cot^{4} (x) - cot^{2} (x) - 15 = 0

cos (θ) = \frac{11}{6}

1 + cos (x) = 3 cos (x)

What is the general solution for cot(x)(1-cos^2(x))= 1/2 sin^2(x) ?
The general solution for cot(x)(1-cos^2(x))= 1/2 sin^2(x) is x=1.10714…+pin