What is the general solution for arccot(x-2)=arccot(x-1)+arccot(x) ?

The general solution for arccot(x-2)=arccot(x-1)+arccot(x) is No Solution for x\in\mathbb{R}

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arccot(x-2)=arccot(x-1)+arccot(x)

Solution

arccot (x - 2) = arccot (x - 1) + arccot (x)

Solution

No Solution for x \in R

Solution steps

arccot (x - 2) = arccot (x - 1) + arccot (x)

Subtract

arccot (x - 1) + arccot (x)

from both sides

arccot (x - 2) - arccot (x - 1) - arccot (x) = 0

Rewrite using trig identities

- arccot (- 1 + x) + arccot (- 2 + x) - arccot (x)

Use the Sum to Product identity:

arccot (s) - arccot (t) = arccot (\frac{s t + 1}{t - s})

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

\frac{( - 2 + x ) ( - 1 + x ) + 1}{- 1 + x - ( - 2 + x )} = x^{2} - 3 x + 3

\frac{( - 2 + x ) ( - 1 + x ) + 1}{- 1 + x - ( - 2 + x )}

Expand

- 1 + x - (- 2 + x) : 1

- 1 + x - (- 2 + x)

- (- 2 + x) : 2 - x

- (- 2 + x)

Distribute parentheses

= - (- 2) - (x)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= 2 - x

= - 1 + x + 2 - x

Simplify

- 1 + x + 2 - x : 1

- 1 + x + 2 - x

Group like terms

= x - x - 1 + 2

Add similar elements:

x - x = 0

= - 1 + 2

Add/Subtract the numbers:

- 1 + 2 = 1

= 1

= 1

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

Expand

(- 2 + x) (- 1 + x) + 1 : x^{2} - 3 x + 3

(- 2 + x) (- 1 + x) + 1

Expand

(- 2 + x) (- 1 + x) : 2 - 3 x + x^{2}

(- 2 + x) (- 1 + x)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = - 2, b = x, c = - 1, d = x

= (- 2) (- 1) + (- 2) x + x (- 1) + xx

Apply minus-plus rules

(- a) (- b) = ab, + (- a) = - a

= 2 \cdot 1 - 2 x - 1 \cdot x + xx

Simplify

2 \cdot 1 - 2 x - 1 \cdot x + xx : 2 - 3 x + x^{2}

2 \cdot 1 - 2 x - 1 \cdot x + xx

Add similar elements:

- 2 x - 1 \cdot x = - 3 x

= 2 \cdot 1 - 3 x + xx

2 \cdot 1 = 2

2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2

xx = x^{2}

xx

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

xx = x^{1 + 1}

= x^{1 + 1}

Add the numbers:

1 + 1 = 2

= x^{2}

= 2 - 3 x + x^{2}

= 2 - 3 x + x^{2}

= 2 - 3 x + x^{2} + 1

Simplify

2 - 3 x + x^{2} + 1 : x^{2} - 3 x + 3

2 - 3 x + x^{2} + 1

Group like terms

= x^{2} - 3 x + 2 + 1

Add the numbers:

2 + 1 = 3

= x^{2} - 3 x + 3

= x^{2} - 3 x + 3

= \frac{x ^{2} - 3 x + 3}{1}

Apply rule

\frac{a}{1} = a

= x^{2} - 3 x + 3

= - arccot (x) + arccot (x^{2} - 3 x + 3)

Use the Sum to Product identity:

arccot (s) - arccot (t) = arccot (\frac{s t + 1}{t - s})

= arccot (\frac{( 3 + x ^{2} - 3 x ) x + 1}{x - ( 3 + x ^{2} - 3 x )})

arccot (\frac{( 3 + x ^{2} - 3 x ) x + 1}{x - ( 3 + x ^{2} - 3 x )}) = 0

0 < arccot (x) < π

No Solution

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

arccot (x - 2) = arccot (x - 1) + arccot (x)

Remove the ones that don't agree with the equation.

No Solution for x \in R

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Frequently Asked Questions (FAQ)

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