What is the general solution for (cot(x))/(cos(x))+1/(sin(x))=1 ?

The general solution for (cot(x))/(cos(x))+1/(sin(x))=1 is No Solution for x\in\mathbb{R}

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(cot(x))/(cos(x))+1/(sin(x))=1

\frac{cot ( x )}{cos ( x )} + \frac{1}{sin ( x )} = 1

No Solution for x \in R

Solution steps

\frac{cot ( x )}{cos ( x )} + \frac{1}{sin ( x )} = 1

Subtract

1

from both sides

\frac{cot ( x )}{cos ( x )} + \frac{1}{sin ( x )} - 1 = 0

Simplify

\frac{cot ( x )}{cos ( x )} + \frac{1}{sin ( x )} - 1 : \frac{cot ( x ) sin ( x ) + cos ( x ) - cos ( x ) sin ( x )}{cos ( x ) sin ( x )}

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

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

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

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

sin (x) - 2 = 0 :

No Solution

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

3 sin (θ) = sin (θ) - 2

sec^{2} (x) - 2 sec (x) + 1 = 0

cos (θ) = sin (θ - \frac{π}{3})

[sin (x) + \frac{1}{2}]^{2} = \frac{1}{2} sin (x) + \frac{13}{16}

- csc (x) + 2 csc^{3} (x) = 0

What is the general solution for (cot(x))/(cos(x))+1/(sin(x))=1 ?
The general solution for (cot(x))/(cos(x))+1/(sin(x))=1 is No Solution for x\in\mathbb{R}