csc^2(x)-2csc(x)=2-4sin(x),0<= x<= 2pi

Solution

csc^{2} (x) - 2 csc (x) = 2 - 4 sin (x), 0 \leq x \leq 2 π

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

Solution steps

csc^{2} (x) - 2 csc (x) = 2 - 4 sin (x), 0 \leq x \leq 2 π

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, 0 \leq x \leq 2 π : x = \frac{π}{6}, x = \frac{5 π}{6}

csc (x) = - 2, 0 \leq x \leq 2 π : x = \frac{5 π}{4}, x = \frac{7 π}{4}

csc (x) = 2, 0 \leq x \leq 2 π : x = \frac{π}{4}, x = \frac{3 π}{4}

Combine all the solutions

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

sin^{2} (x) = \frac{( 9 m - 4 )}{7}

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

6 sin^{2} (θ) - sin (θ) - 1 = 0

cos (x) = tan (x) sin (x)

1 - 2 cos^{2} (8 x) = sin (4 x)