csc^2(x)=4cot^2(x)

Solution

csc^{2} (x) = 4 cot^{2} (x)

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Solution steps

csc^{2} (x) = 4 cot^{2} (x)

Subtract

4 cot^{2} (x)

from both sides

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

Factor

csc^{2} (x) - 4 cot^{2} (x) : (csc (x) + 2 cot (x)) (csc (x) - 2 cot (x))

(csc (x) + 2 cot (x)) (csc (x) - 2 cot (x)) = 0

Solving each part separately

csc (x) + 2 cot (x) = 0 or csc (x) - 2 cot (x) = 0

csc (x) + 2 cot (x) = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

csc (x) - 2 cot (x) = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

x, cos (x) = 10

3 cos (θ) = 2, 1

3 csc (x) - 7 = 0

tan (2 θ) = 0.6583