2sin^4(x)=cos^2(x)

Solution

2 sin^{4} (x) = cos^{2} (x)

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Solution steps

2 sin^{4} (x) = cos^{2} (x)

Subtract

cos^{2} (x)

from both sides

2 sin^{4} (x) - cos^{2} (x) = 0

Factor

2 sin^{4} (x) - cos^{2} (x) : (2 sin^{2} (x) + cos (x)) (2 sin^{2} (x) - cos (x))

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

Solving each part separately

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

2 sin^{2} (x) + cos (x) = 0 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

2 sin^{2} (x) - cos (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

2 cos (3 x) = cot (3 x)

cos (x) (cos (x) - 1) = 0, n = 0

1 + cos^{2} (x) = 0

tan (2 x) = - 1.238

tan (θ) = π