What is the general solution for arctan(x+1)+arctan(x-1)=arctan(8/31) ?

The general solution for arctan(x+1)+arctan(x-1)=arctan(8/31) is x= 1/4

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arctan(x+1)+arctan(x-1)=arctan(8/31)

Solution

arctan (x + 1) + arctan (x - 1) = arctan (\frac{8}{31})

Solution

x = \frac{1}{4}

Solution steps

arctan (x + 1) + arctan (x - 1) = arctan (\frac{8}{31})

Rewrite using trig identities

arctan (x + 1) + arctan (x - 1)

Use the Sum to Product identity:

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

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

arctan (\frac{x + 1 + x - 1}{1 - ( x + 1 ) ( x - 1 )}) = arctan (\frac{8}{31})

Apply trig inverse properties

arctan (\frac{x + 1 + x - 1}{1 - ( x + 1 ) ( x - 1 )}) = arctan (\frac{8}{31})

arctan (x) = a \Rightarrow x = tan (a)

\frac{x + 1 + x - 1}{1 - ( x + 1 ) ( x - 1 )} = tan (arctan (\frac{8}{31}))

tan (arctan (\frac{8}{31})) = \frac{8}{31}

tan (arctan (\frac{8}{31}))

Rewrite using trig identities:

tan (arctan (\frac{8}{31})) = \frac{8}{31}

Use the following identity:

tan (arctan (x)) = x

= \frac{8}{31}

= \frac{8}{31}

\frac{x + 1 + x - 1}{1 - ( x + 1 ) ( x - 1 )} = \frac{8}{31}

\frac{x + 1 + x - 1}{1 - ( x + 1 ) ( x - 1 )} = \frac{8}{31}

Solve

\frac{x + 1 + x - 1}{1 - ( x + 1 ) ( x - 1 )} = \frac{8}{31} : x = - 8, x = \frac{1}{4}

\frac{x + 1 + x - 1}{1 - ( x + 1 ) ( x - 1 )} = \frac{8}{31}

Cross multiply

\frac{x + 1 + x - 1}{1 - ( x + 1 ) ( x - 1 )} = \frac{8}{31}

Simplify

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

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

x + 1 + x - 1 = 2 x

x + 1 + x - 1

Group like terms

= x + x + 1 - 1

Add similar elements:

x + x = 2 x

= 2 x + 1 - 1

1 - 1 = 0

= 2 x

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

Expand

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

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

Expand

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

Expand

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

(x + 1) (x - 1)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = x, b = 1

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

Apply rule

1^{a} = 1

1^{2} = 1

= x^{2} - 1

= - (x^{2} - 1)

Distribute parentheses

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

Apply minus-plus rules

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

= - x^{2} + 1

= 1 - x^{2} + 1

Simplify

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

1 - x^{2} + 1

Group like terms

= - x^{2} + 1 + 1

Add the numbers:

1 + 1 = 2

= - x^{2} + 2

= - x^{2} + 2

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

\frac{2 x}{- x ^{2} + 2} = \frac{8}{31}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

2 x \cdot 31 = (- x^{2} + 2) \cdot 8

Simplify

2 x \cdot 31 : 62 x

2 x \cdot 31

Multiply the numbers:

2 \cdot 31 = 62

= 62 x

62 x = (- x^{2} + 2) \cdot 8

62 x = (- x^{2} + 2) \cdot 8

Solve

62 x = (- x^{2} + 2) \cdot 8 : x = - 8, x = \frac{1}{4}

62 x = (- x^{2} + 2) \cdot 8

Expand

(- x^{2} + 2) \cdot 8 : - 8 x^{2} + 16

(- x^{2} + 2) \cdot 8

= 8 (- x^{2} + 2)

Apply the distributive law:

a (b + c) = ab + a c

a = 8, b = - x^{2}, c = 2

= 8 (- x^{2}) + 8 \cdot 2

Apply minus-plus rules

+ (- a) = - a

= - 8 x^{2} + 8 \cdot 2

Multiply the numbers:

8 \cdot 2 = 16

= - 8 x^{2} + 16

62 x = - 8 x^{2} + 16

Switch sides

- 8 x^{2} + 16 = 62 x

Move

62 x

to the left side

- 8 x^{2} + 16 = 62 x

Subtract

62 x

from both sides

- 8 x^{2} + 16 - 62 x = 62 x - 62 x

Simplify

- 8 x^{2} + 16 - 62 x = 0

- 8 x^{2} + 16 - 62 x = 0

Write in the standard form

a x^{2} + b x + c = 0

- 8 x^{2} - 62 x + 16 = 0

Solve with the quadratic formula

- 8 x^{2} - 62 x + 16 = 0

Quadratic Equation Formula:

For

a = - 8, b = - 62, c = 16

x_{1, 2} = \frac{- ( - 62 ) \pm ( - 62 ) ^{2} - 4 ( - 8 ) \cdot 16}{2 ( - 8 )}

x_{1, 2} = \frac{- ( - 62 ) \pm ( - 62 ) ^{2} - 4 ( - 8 ) \cdot 16}{2 ( - 8 )}

(- 62)^{2} - 4 (- 8) \cdot 16 = 66

(- 62)^{2} - 4 (- 8) \cdot 16

Apply rule

- (- a) = a

= (- 62)^{2} + 4 \cdot 8 \cdot 16

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 62)^{2} = 6 2^{2}

= 6 2^{2} + 4 \cdot 8 \cdot 16

Multiply the numbers:

4 \cdot 8 \cdot 16 = 512

= 6 2^{2} + 512

6 2^{2} = 3844

= 3844 + 512

Add the numbers:

3844 + 512 = 4356

= 4356

Factor the number:

4356 = 6 6^{2}

= 6 6^{2}

Apply radical rule:

6 6^{2} = 66

= 66

x_{1, 2} = \frac{- ( - 62 ) \pm 66}{2 ( - 8 )}

Separate the solutions

x_{1} = \frac{- ( - 62 ) + 66}{2 ( - 8 )}, x_{2} = \frac{- ( - 62 ) - 66}{2 ( - 8 )}

x = \frac{- ( - 62 ) + 66}{2 ( - 8 )} : - 8

\frac{- ( - 62 ) + 66}{2 ( - 8 )}

Remove parentheses:

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

= \frac{62 + 66}{- 2 \cdot 8}

Add the numbers:

62 + 66 = 128

= \frac{128}{- 2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{128}{- 16}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{128}{16}

Divide the numbers:

\frac{128}{16} = 8

= - 8

x = \frac{- ( - 62 ) - 66}{2 ( - 8 )} : \frac{1}{4}

\frac{- ( - 62 ) - 66}{2 ( - 8 )}

Remove parentheses:

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

= \frac{62 - 66}{- 2 \cdot 8}

Subtract the numbers:

62 - 66 = - 4

= \frac{- 4}{- 2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{- 4}{- 16}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4}{16}

Cancel the common factor:

4

= \frac{1}{4}

The solutions to the quadratic equation are:

x = - 8, x = \frac{1}{4}

x = - 8, x = \frac{1}{4}

Verify Solutions

Find undefined (singularity) points:

x = - 2, x = 2

Take the denominator(s) of

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

and compare to zero

Solve

1 - (x + 1) (x - 1) = 0 : x = - 2, x = 2

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

Expand

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

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

Expand

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

Expand

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

(x + 1) (x - 1)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = x, b = 1

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

Apply rule

1^{a} = 1

1^{2} = 1

= x^{2} - 1

= - (x^{2} - 1)

Distribute parentheses

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

Apply minus-plus rules

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

= - x^{2} + 1

= 1 - x^{2} + 1

Simplify

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

1 - x^{2} + 1

Group like terms

= - x^{2} + 1 + 1

Add the numbers:

1 + 1 = 2

= - x^{2} + 2

= - x^{2} + 2

- x^{2} + 2 = 0

Solve with the quadratic formula

- x^{2} + 2 = 0

Quadratic Equation Formula:

For

a = - 1, b = 0, c = 2

x_{1, 2} = \frac{- 0 \pm 0 ^{2} - 4 ( - 1 ) \cdot 2}{2 ( - 1 )}

x_{1, 2} = \frac{- 0 \pm 0 ^{2} - 4 ( - 1 ) \cdot 2}{2 ( - 1 )}

0^{2} - 4 (- 1) \cdot 2 = 22

0^{2} - 4 (- 1) \cdot 2

Apply rule

0^{a} = 0

0^{2} = 0

= 0 - 4 (- 1) \cdot 2

Apply rule

- (- a) = a

= 0 + 4 \cdot 1 \cdot 2

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 0 + 8

Add the numbers:

0 + 8 = 8

= 8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

x_{1, 2} = \frac{- 0 \pm 2 2}{2 ( - 1 )}

Separate the solutions

x_{1} = \frac{- 0 + 2 2}{2 ( - 1 )}, x_{2} = \frac{- 0 - 2 2}{2 ( - 1 )}

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

\frac{- 0 + 2 2}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 0 + 2 2}{- 2 \cdot 1}

- 0 + 22 = 22

= \frac{2 2}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 2}{- 2}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2 2}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 2

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

\frac{- 0 - 2 2}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 0 - 2 2}{- 2 \cdot 1}

- 0 - 22 = - 22

= \frac{- 2 2}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2 2}{- 2}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{2 2}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2

The solutions to the quadratic equation are:

x = - 2, x = 2

The following points are undefined

x = - 2, x = 2

Combine undefined points with solutions:

x = - 8, x = \frac{1}{4}

x = - 8, x = \frac{1}{4}

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

arctan (x + 1) + arctan (x - 1) = arctan (\frac{8}{31})

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

Check the solution

- 8 :

False

- 8

Plug in

n = 1

- 8

For

arctan (x + 1) + arctan (x - 1) = arctan (\frac{8}{31})

plug in

x = - 8

arctan (- 8 + 1) + arctan (- 8 - 1) = arctan (\frac{8}{31})

Refine

- 2.88903 \dots = 0.25255 \dots

\Rightarrow False

Check the solution

\frac{1}{4} :

True

\frac{1}{4}

Plug in

n = 1

\frac{1}{4}

For

arctan (x + 1) + arctan (x - 1) = arctan (\frac{8}{31})

plug in

x = \frac{1}{4}

arctan (\frac{1}{4} + 1) + arctan (\frac{1}{4} - 1) = arctan (\frac{8}{31})

Refine

0.25255 \dots = 0.25255 \dots

\Rightarrow True

x = \frac{1}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for arctan(x+1)+arctan(x-1)=arctan(8/31) ?
The general solution for arctan(x+1)+arctan(x-1)=arctan(8/31) is x= 1/4