What is the general solution for 4cos^2(x)-4sin(x)-1=0 ?

The general solution for 4cos^2(x)-4sin(x)-1=0 is x= pi/6+2pin,x=(5pi)/6+2pin

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

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

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

Solution steps

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

Rewrite using trig identities

3 - 4 sin (x) - 4 sin^{2} (x) = 0

Solve by substitution

sin (x) = - \frac{3}{2}, sin (x) = \frac{1}{2}

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

No Solution

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

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

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

sin^{3} (x) - 2 sin (x) = 0

2 cos^{3} (x) = cot^{3} (x)

\frac{1}{( sec ^{2} ( a ) )} + \frac{1}{( cos ^{2} ( a ) )} = 1

(1 - cos (a)) (1 + cos (a)) = tan (a) sin (a)

What is the general solution for 4cos^2(x)-4sin(x)-1=0 ?
The general solution for 4cos^2(x)-4sin(x)-1=0 is x= pi/6+2pin,x=(5pi)/6+2pin