What is the general solution for (sec(x)+csc(x))/(1+tan(x))=sin(x) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(sec(x)+csc(x))/(1+tan(x))=sin(x)

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} = sin (x)

No Solution for x \in R

Solution steps

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} = sin (x)

Subtract

sin (x)

from both sides

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} - sin (x) = 0

Simplify

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} - sin (x) : \frac{sec ( x ) + csc ( x ) - sin ( x ) ( 1 + tan ( x ) )}{1 + tan ( x )}

\frac{sec ( x ) + csc ( x ) - sin ( x ) ( 1 + tan ( x ) )}{1 + tan ( x )} = 0

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

sec (x) + csc (x) - sin (x) (1 + tan (x)) = 0

Express with sin, cos

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

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

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

Factor

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

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

Solving each part separately

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

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

sin (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn

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

Combine all the solutions

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

Since the equation is undefined for:

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

No Solution for x \in R

sec (x) - \frac{sin ( x )}{cos ( x )} = cos (x)

- 10 sin^{2} (t) = 0

x, e^{y} cos (2 x) + y = 1

cos (θ) = 0.4, sec (θ)

- \frac{π}{2} sin (\frac{π x}{2}) = 0

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