What is the general solution for (sin(x))/(cos(x))=(sin(x))/(1+cot^2(x)) ?

The general solution for (sin(x))/(cos(x))=(sin(x))/(1+cot^2(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 >

(sin(x))/(cos(x))=(sin(x))/(1+cot^2(x))

\frac{sin ( x )}{cos ( x )} = \frac{sin ( x )}{1 + cot ^{2} ( x )}

No Solution for x \in R

Solution steps

\frac{sin ( x )}{cos ( x )} = \frac{sin ( x )}{1 + cot ^{2} ( x )}

Subtract

\frac{sin ( x )}{1 + cot ^{2} ( x )}

from both sides

\frac{sin ( x )}{cos ( x )} - \frac{sin ( x )}{1 + cot ^{2} ( x )} = 0

Simplify

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

\frac{sin ( x ) ( cot ^{2} ( x ) + 1 ) - sin ( x ) cos ( x )}{cos ( x ) ( cot ^{2} ( x ) + 1 )} = 0

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

sin (x) (cot^{2} (x) + 1) - sin (x) cos (x) = 0

Rewrite using trig identities

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

Factor

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

sin (x) (1 + \frac{cos ^{2} ( x )}{sin ^{2} ( x )} - cos (x)) = 0

Solving each part separately

sin (x) = 0 or 1 + \frac{cos ^{2} ( x )}{sin ^{2} ( x )} - cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

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

No Solution

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

3 + 8 cos (2 θ) = - 1

- 4 cos^{2} (x) + 6 cos (x) = 2

sec (θ) = - \frac{4}{3}

tan (2 x) = tan (201 8^{\circ})

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

What is the general solution for (sin(x))/(cos(x))=(sin(x))/(1+cot^2(x)) ?
The general solution for (sin(x))/(cos(x))=(sin(x))/(1+cot^2(x)) is No Solution for x\in\mathbb{R}