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

The general solution for 0=cos(x)-sin(2x) is x= pi/2+2pin,x=(3pi)/2+2pin,x= pi/6+2pin,x=(5pi)/6+2pin

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0=cos(x)-sin(2x)

0 = cos (x) - sin (2 x)

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

Solution steps

0 = cos (x) - sin (2 x)

Switch sides

cos (x) - sin (2 x) = 0

Rewrite using trig identities

cos (x) - 2 cos (x) sin (x) = 0

Factor

cos (x) - 2 cos (x) sin (x) : - cos (x) (2 sin (x) - 1)

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

Solving each part separately

cos (x) = 0 or 2 sin (x) - 1 = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

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

tan^{2} (x) + 2 + 2 tan (x) - 7 = 0

(2 cos (x) + 1) = 0

3 cos (x) + 2 = 0

2 tan (x) sin (x) - tan (x) = 0

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

What is the general solution for 0=cos(x)-sin(2x) ?
The general solution for 0=cos(x)-sin(2x) is x= pi/2+2pin,x=(3pi)/2+2pin,x= pi/6+2pin,x=(5pi)/6+2pin