What is the general solution for (sin^2(x))/4 =(cos^2(x))/2 ?

The general solution for (sin^2(x))/4 =(cos^2(x))/2 is x=-0.95531…+pin,x=0.95531…+pin

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

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

x = - 0.95531 \dots + πn, x = 0.95531 \dots + πn

Solution steps

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

Subtract

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

from both sides

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

Simplify

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

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

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

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

Factor

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

(sin (x) + 2 cos (x)) (sin (x) - 2 cos (x)) = 0

Solving each part separately

sin (x) + 2 cos (x) = 0 or sin (x) - 2 cos (x) = 0

sin (x) + 2 cos (x) = 0 : x = arctan (- 2) + πn

sin (x) - 2 cos (x) = 0 : x = arctan (2) + πn

Combine all the solutions

x = arctan (- 2) + πn, x = arctan (2) + πn

Show solutions in decimal form

x = - 0.95531 \dots + πn, x = 0.95531 \dots + πn

θ, 2 sin (θ) = tan (θ)

5 cot^{2} (x) + 8 cot (x) + 3 = 0

- 2 sin^{2} (x) - cos (x) + 3 = 0

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

sin (x) = \frac{15}{31}

What is the general solution for (sin^2(x))/4 =(cos^2(x))/2 ?
The general solution for (sin^2(x))/4 =(cos^2(x))/2 is x=-0.95531…+pin,x=0.95531…+pin