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

Solution

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

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

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

Subtract

\frac{csc ^{2} ( x )}{4}

from both sides

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

Simplify

4 sin^{2} (x) - \frac{csc ^{2} ( x )}{4} : \frac{16 sin ^{2} ( x ) - csc ^{2} ( x )}{4}

\frac{16 sin ^{2} ( x ) - csc ^{2} ( x )}{4} = 0

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

16 sin^{2} (x) - csc^{2} (x) = 0

Factor

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

(4 sin (x) + csc (x)) (4 sin (x) - csc (x)) = 0

Solving each part separately

4 sin (x) + csc (x) = 0 or 4 sin (x) - csc (x) = 0

4 sin (x) + csc (x) = 0 :

No Solution

4 sin (x) - csc (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

2 sin (3 x) = 2

sin (x) = \frac{2}{3}

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

cos (x) - sin (x) = \frac{2}{3}

0 = ((10 cos (x)) - 8)^{2} ((10 sin (x)) - 6)^{2}