solvefor x,sin^4(x)=(cos^2(x))/2

Solution

solve for

x, sin^{4} (x) = \frac{cos ^{2} ( x )}{2}

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

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

Subtract

\frac{cos ^{2} ( x )}{2}

from both sides

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

Simplify

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

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

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

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

tan^{2} (x) + 5 tan (x) - 1 = 0

csc (θ) + 1 = 0

2 sin^{2} (θ) - 3 sin (θ) + 1 = 0, 0 \leq θ \leq 2 π

2 sec (x) + 2 tan (x) = 2

2 sin^{2} (θ) + 3 sin (θ) = 0