What is the general solution for sqrt(2)sin(x)sec(x)=2sin(x) ?

The general solution for sqrt(2)sin(x)sec(x)=2sin(x) is x=2pin,x=pi+2pin,x= pi/4+2pin,x=(7pi)/4+2pin

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sqrt(2)sin(x)sec(x)=2sin(x)

2 sin (x) sec (x) = 2 sin (x)

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

Solution steps

2 sin (x) sec (x) = 2 sin (x)

Subtract

2 sin (x)

from both sides

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

Factor

2 sin (x) sec (x) - 2 sin (x) : sin (x) (2 sec (x) - 2)

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

Solving each part separately

sin (x) = 0 or 2 sec (x) - 2 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

2 sec (x) - 2 = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

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

tan^{3} (x) = tan (x)

3 - 2 sin (x) = 0

- 9 cos^{2} (θ) + 14 sin (θ) + 14 = 0

cos (x) tan (x) = - 7 tan (x)

12 sin^{2} (θ) - 9 = 0

What is the general solution for sqrt(2)sin(x)sec(x)=2sin(x) ?
The general solution for sqrt(2)sin(x)sec(x)=2sin(x) is x=2pin,x=pi+2pin,x= pi/4+2pin,x=(7pi)/4+2pin