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

The general solution for sin(2x)sec(x)+2cos(x)=0 is x=(3pi)/4+pin

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sin(2x)sec(x)+2cos(x)=0

sin (2 x) sec (x) + 2 cos (x) = 0

x = \frac{3 π}{4} + πn

Solution steps

sin (2 x) sec (x) + 2 cos (x) = 0

Rewrite using trig identities

2 cos (x) + 2 cos (x) sec (x) sin (x) = 0

Factor

2 cos (x) + 2 cos (x) sec (x) sin (x) : 2 cos (x) (1 + sec (x) sin (x))

2 cos (x) (1 + sec (x) sin (x)) = 0

Solving each part separately

cos (x) = 0 or 1 + sec (x) sin (x) = 0

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

1 + sec (x) sin (x) = 0 : x = \frac{3 π}{4} + πn

Combine all the solutions

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

Since the equation is undefined for:

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn

x = \frac{3 π}{4} + πn

3 + 2 sin (x) = 0

tan (x) = \frac{6}{4}

cot (x) sin^{2} (x) = 4 cot (x)

x, sin (x) + cos (y) = sin (x) cos (y)

x, z = arctan (\frac{x}{y})

What is the general solution for sin(2x)sec(x)+2cos(x)=0 ?
The general solution for sin(2x)sec(x)+2cos(x)=0 is x=(3pi)/4+pin