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

Solution

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

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Solution steps

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

Subtract

2 - 4 sin (x)

from both sides

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

Rewrite using trig identities

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

Solve by substitution

csc (x) = 2, csc (x) = - 2, csc (x) = 2

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

csc (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

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

cos (θ) = - \frac{1}{2}

3 sin (2 t) + 4 = 1

x, cos (x) = 0

cos (x + \frac{π}{6}) - cos (x - \frac{π}{6}) = 1