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

The general solution for (sin(2x))/(cos(x))=2 is No Solution for x\in\mathbb{R}

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(sin(2x))/(cos(x))=2

\frac{sin ( 2 x )}{cos ( x )} = 2

No Solution for x \in R

Solution steps

\frac{sin ( 2 x )}{cos ( x )} = 2

Subtract

2

from both sides

\frac{sin ( 2 x )}{cos ( x )} - 2 = 0

Simplify

\frac{sin ( 2 x )}{cos ( x )} - 2 : \frac{sin ( 2 x ) - 2 cos ( x )}{cos ( x )}

\frac{sin ( 2 x ) - 2 cos ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

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

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

Combine all the solutions

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

Since the equation is undefined for:

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

No Solution for x \in R

sin (x) = 0.8

6 cos (2 x) = 0

sin (x) = 1 - cos (x)

3 cos (x) = 0

csc (x) = \frac{2 3}{3}

What is the general solution for (sin(2x))/(cos(x))=2 ?
The general solution for (sin(2x))/(cos(x))=2 is No Solution for x\in\mathbb{R}