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

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

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(tan(x))/(sin(x))=cos(x)

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

No Solution for x \in R

Solution steps

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

Subtract

cos (x)

from both sides

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

Simplify

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

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

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

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

Express with sin, cos

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

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

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

Factor

sin (x) - cos^{2} (x) sin (x) : - sin (x) (cos (x) + 1) (cos (x) - 1)

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

cos (x) + 1 = 0 : x = π + 2 πn

cos (x) - 1 = 0 : x = 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

No Solution for x \in R

4 tan^{2} (x) + 7 tan (x) - 4 = 0

tan (θ) = - 1 - 3

y, v = (1 - 4 cos (5 y))^{- \frac{1}{2}}

x, cos (x) = cos (p)

cot (a) - 1 = 0

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