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

The general solution for cot^2(x)-cos^2(x)=-1 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 >

cot^2(x)-cos^2(x)=-1

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

- 1

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

1 + cot^{2} (x) = csc^{2} (x)

= - cos^{2} (x) + csc^{2} (x)

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

Factor

- cos^{2} (x) + csc^{2} (x) : (csc (x) + cos (x)) (csc (x) - cos (x))

- cos^{2} (x) + csc^{2} (x)

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

csc^{2} (x) - cos^{2} (x) = (csc (x) + cos (x)) (csc (x) - cos (x))

= (csc (x) + cos (x)) (csc (x) - cos (x))

(csc (x) + cos (x)) (csc (x) - cos (x)) = 0

Solving each part separately

csc (x) + cos (x) = 0 or csc (x) - cos (x) = 0

csc (x) + cos (x) = 0 :

No Solution

csc (x) + cos (x) = 0

Express with sin, cos

cos (x) + csc (x)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

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

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

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

Rewrite using trig identities

1 + cos (x) sin (x)

Use the Double Angle identity:

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

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

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

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

\frac{sin ( 2 x )}{2} = - 1

\frac{sin ( 2 x )}{2} = - 1

Multiply both sides by

2

\frac{sin ( 2 x )}{2} = - 1

Multiply both sides by

2

\frac{2 sin ( 2 x )}{2} = 2 (- 1)

Simplify

sin (2 x) = - 2

sin (2 x) = - 2

- 1 \leq sin (x) \leq 1

No Solution

csc (x) - cos (x) = 0 :

No Solution

csc (x) - cos (x) = 0

Express with sin, cos

- cos (x) + csc (x)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

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

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

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

Rewrite using trig identities

1 - cos (x) sin (x)

Use the Double Angle identity:

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

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

= 1 - \frac{sin ( 2 x )}{2}

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

- \frac{sin ( 2 x )}{2} = - 1

- \frac{sin ( 2 x )}{2} = - 1

Multiply both sides by

2

- \frac{sin ( 2 x )}{2} = - 1

Multiply both sides by

2

2 (- \frac{sin ( 2 x )}{2}) = 2 (- 1)

Simplify

- sin (2 x) = - 2

- sin (2 x) = - 2

Divide both sides by

- 1

- sin (2 x) = - 2

Divide both sides by

- 1

\frac{- sin ( 2 x )}{- 1} = \frac{- 2}{- 1}

Simplify

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

Popular Examples

sin^2(θ)=2*sin(θ)+3

sin^{2} (θ) = 2 \cdot sin (θ) + 3

tan(3x)=1,-2pi<= x<= 2pi

tan (3 x) = 1, - 2 π \leq x \leq 2 π

sin(x)=(-3)/2

sin (x) = \frac{- 3}{2}

cos(x)sin(2x)+3sin^2(x)=-3sin(x)

cos (x) sin (2 x) + 3 sin^{2} (x) = - 3 sin (x)

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

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

Frequently Asked Questions (FAQ)

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