(4sin^2(x)-1)(cos^2(x)-1)=0

Solution

(4 sin^{2} (x) - 1) (cos^{2} (x) - 1) = 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, x = 2 πn, x = π + 2 πn

Solution steps

(4 sin^{2} (x) - 1) (cos^{2} (x) - 1) = 0

Solving each part separately

4 sin^{2} (x) - 1 = 0 or cos^{2} (x) - 1 = 0

4 sin^{2} (x) - 1 = 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

cos^{2} (x) - 1 = 0 : x = 2 πn, x = π + 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, x = 2 πn, x = π + 2 πn

3 sin^{2} (x) - 6 sin (x) = - 3, 0 \leq x \leq 2 π

1 + sin (π \cdot t) = \frac{3}{2}

- cos (x) = sin^{2} (x) - 1

csc (x) = \frac{5}{2}

csc (x) = \frac{5}{4}