4cos^2(x)+2cos(x)-2sqrt(2)cos(x)=sqrt(2)

Solution

4 cos^{2} (x) + 2 cos (x) - 22 cos (x) = 2

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

Solution steps

4 cos^{2} (x) + 2 cos (x) - 22 cos (x) = 2

Solve by substitution

cos (x) = \frac{2}{2}, cos (x) = - \frac{1}{2}

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

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Combine all the solutions

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

sin (x + 2 0^{\circ}) \cdot csc (7 0^{\circ}) = 1

\frac{π}{2} = \frac{arcsin ( x )}{c}

6 sin^{2} (x) + 16 sin (x) cos (x) - 6 cos^{2} (x) = 0

sec^{2} (t) = 0

x, t = tan (\frac{x}{2})