What is the general solution for arctan(x/3)+arctan(x/2)=arctan(x) ?

The general solution for arctan(x/3)+arctan(x/2)=arctan(x) is x=0,x=-1,x=1

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arctan(x/3)+arctan(x/2)=arctan(x)

Solution

arctan (\frac{x}{3}) + arctan (\frac{x}{2}) = arctan (x)

Solution

x = 0, x = - 1, x = 1

Solution steps

arctan (\frac{x}{3}) + arctan (\frac{x}{2}) = arctan (x)

Subtract

arctan (x)

from both sides

arctan (\frac{x}{3}) + arctan (\frac{x}{2}) - arctan (x) = 0

Rewrite using trig identities

- arctan (x) + arctan (\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}})

Use the Sum to Product identity:

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

= arctan \frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x}

arctan \frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = 0

Apply trig inverse properties

arctan \frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = 0

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

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = tan (0)

tan (0) = 0

tan (0)

Use the following trivial identity:

tan (0) = 0

tan (0)

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 0

= 0

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = 0

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = 0

Solve

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = 0 : x = 0, x = - 1, x = 1

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = 0

Simplify

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} : \frac{- x + x ^{3}}{6 + 4 x ^{2}}

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x}

\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x = \frac{5 x ^{2}}{6 - x ^{2}}

\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x

\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} = \frac{5 x}{6 \cdot \frac{6 - x ^{2}}{6}}

\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}}

\frac{x}{3} \cdot \frac{x}{2} = \frac{x ^{2}}{6}

\frac{x}{3} \cdot \frac{x}{2}

Multiply fractions:

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

= \frac{xx}{3 \cdot 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}

= \frac{x ^{2}}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{x ^{2}}{6}

= \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x ^{2}}{6}}

Join

\frac{x}{3} + \frac{x}{2} : \frac{5 x}{6}

\frac{x}{3} + \frac{x}{2}

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{x}{3} :

multiply the denominator and numerator by

2

\frac{x}{3} = \frac{x \cdot 2}{3 \cdot 2} = \frac{x \cdot 2}{6}

For

\frac{x}{2} :

multiply the denominator and numerator by

3

\frac{x}{2} = \frac{x \cdot 3}{2 \cdot 3} = \frac{x \cdot 3}{6}

= \frac{x \cdot 2}{6} + \frac{x \cdot 3}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 2 + x \cdot 3}{6}

Add similar elements:

2 x + 3 x = 5 x

= \frac{5 x}{6}

= \frac{\frac{5 x}{6}}{1 - \frac{x ^{2}}{6}}

Apply the fraction rule:

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

= \frac{5 x}{6 ( 1 - \frac{x ^{2}}{6} )}

Join

1 - \frac{x ^{2}}{6} : \frac{6 - x ^{2}}{6}

1 - \frac{x ^{2}}{6}

Convert element to fraction:

1 = \frac{1 \cdot 6}{6}

= \frac{1 \cdot 6}{6} - \frac{x ^{2}}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 6 - x ^{2}}{6}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6 - x ^{2}}{6}

= \frac{5 x}{6 \cdot \frac{- x ^{2} + 6}{6}}

= \frac{5 x}{6 \cdot \frac{- x ^{2} + 6}{6}} x

Multiply fractions:

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

= \frac{5 xx}{6 \cdot \frac{6 - x ^{2}}{6}}

5 xx = 5 x^{2}

5 xx

Apply exponent rule:

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

xx = x^{1 + 1}

= 5 x^{1 + 1}

Add the numbers:

1 + 1 = 2

= 5 x^{2}

= \frac{5 x ^{2}}{6 \cdot \frac{- x ^{2} + 6}{6}}

Multiply

6 \cdot \frac{6 - x ^{2}}{6} : 6 - x^{2}

6 \cdot \frac{6 - x ^{2}}{6}

Multiply fractions:

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

= \frac{( 6 - x ^{2} ) \cdot 6}{6}

Cancel the common factor:

6

= 6 - x^{2}

= \frac{5 x ^{2}}{6 - x ^{2}}

= \frac{\frac{\frac{x}{3} + \frac{x}{2}}{- \frac{x}{3} \cdot \frac{x}{2} + 1} - x}{1 + \frac{5 x ^{2}}{- x ^{2} + 6}}

\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} = \frac{5 x}{6 \cdot \frac{6 - x ^{2}}{6}}

\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}}

\frac{x}{3} \cdot \frac{x}{2} = \frac{x ^{2}}{6}

\frac{x}{3} \cdot \frac{x}{2}

Multiply fractions:

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

= \frac{xx}{3 \cdot 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}

= \frac{x ^{2}}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{x ^{2}}{6}

= \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x ^{2}}{6}}

Join

\frac{x}{3} + \frac{x}{2} : \frac{5 x}{6}

\frac{x}{3} + \frac{x}{2}

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{x}{3} :

multiply the denominator and numerator by

2

\frac{x}{3} = \frac{x \cdot 2}{3 \cdot 2} = \frac{x \cdot 2}{6}

For

\frac{x}{2} :

multiply the denominator and numerator by

3

\frac{x}{2} = \frac{x \cdot 3}{2 \cdot 3} = \frac{x \cdot 3}{6}

= \frac{x \cdot 2}{6} + \frac{x \cdot 3}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 2 + x \cdot 3}{6}

Add similar elements:

2 x + 3 x = 5 x

= \frac{5 x}{6}

= \frac{\frac{5 x}{6}}{1 - \frac{x ^{2}}{6}}

Apply the fraction rule:

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

= \frac{5 x}{6 ( 1 - \frac{x ^{2}}{6} )}

Join

1 - \frac{x ^{2}}{6} : \frac{6 - x ^{2}}{6}

1 - \frac{x ^{2}}{6}

Convert element to fraction:

1 = \frac{1 \cdot 6}{6}

= \frac{1 \cdot 6}{6} - \frac{x ^{2}}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 6 - x ^{2}}{6}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6 - x ^{2}}{6}

= \frac{5 x}{6 \cdot \frac{- x ^{2} + 6}{6}}

= \frac{\frac{5 x}{6 \cdot \frac{- x ^{2} + 6}{6}} - x}{1 + \frac{5 x ^{2}}{- x ^{2} + 6}}

Join

1 + \frac{5 x ^{2}}{6 - x ^{2}} : \frac{6 + 4 x ^{2}}{6 - x ^{2}}

1 + \frac{5 x ^{2}}{6 - x ^{2}}

Convert element to fraction:

1 = \frac{1 ( 6 - x ^{2} )}{6 - x ^{2}}

= \frac{1 \cdot ( 6 - x ^{2} )}{6 - x ^{2}} + \frac{5 x ^{2}}{6 - x ^{2}}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot ( 6 - x ^{2} ) + 5 x ^{2}}{6 - x ^{2}}

1 \cdot (6 - x^{2}) + 5 x^{2} = 6 + 4 x^{2}

1 \cdot (6 - x^{2}) + 5 x^{2}

1 \cdot (6 - x^{2}) = 6 - x^{2}

1 \cdot (6 - x^{2})

Multiply:

1 \cdot (6 - x^{2}) = (6 - x^{2})

= (6 - x^{2})

Remove parentheses:

(a) = a

= 6 - x^{2}

= 6 - x^{2} + 5 x^{2}

Add similar elements:

- x^{2} + 5 x^{2} = 4 x^{2}

= 6 + 4 x^{2}

= \frac{6 + 4 x ^{2}}{6 - x ^{2}}

= \frac{\frac{5 x}{6 \cdot \frac{- x ^{2} + 6}{6}} - x}{\frac{6 + 4 x ^{2}}{6 - x ^{2}}}

Join

\frac{5 x}{6 \cdot \frac{6 - x ^{2}}{6}} - x : \frac{- x + x ^{3}}{6 - x ^{2}}

\frac{5 x}{6 \cdot \frac{6 - x ^{2}}{6}} - x

Convert element to fraction:

x = \frac{x 6 \frac{6 - x ^{2}}{6}}{6 \frac{6 - x ^{2}}{6}}

= \frac{5 x}{6 \cdot \frac{6 - x ^{2}}{6}} - \frac{x \cdot 6 \cdot \frac{6 - x ^{2}}{6}}{6 \cdot \frac{6 - x ^{2}}{6}}

Since the denominators are equal, combine the fractions:

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

= \frac{5 x - x \cdot 6 \cdot \frac{6 - x ^{2}}{6}}{6 \cdot \frac{6 - x ^{2}}{6}}

Multiply

6 \cdot \frac{6 - x ^{2}}{6} : 6 - x^{2}

6 \cdot \frac{6 - x ^{2}}{6}

Multiply fractions:

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

= \frac{( 6 - x ^{2} ) \cdot 6}{6}

Cancel the common factor:

6

= 6 - x^{2}

= \frac{5 x - 6 \cdot \frac{- x ^{2} + 6}{6} x}{6 - x ^{2}}

x \cdot 6 \cdot \frac{6 - x ^{2}}{6} = x (6 - x^{2})

x \cdot 6 \cdot \frac{6 - x ^{2}}{6}

Multiply fractions:

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

= \frac{( 6 - x ^{2} ) x \cdot 6}{6}

Cancel the common factor:

6

= (6 - x^{2}) x

= \frac{5 x - x ( - x ^{2} + 6 )}{6 - x ^{2}}

Expand

5 x - (6 - x^{2}) x : - x + x^{3}

5 x - (6 - x^{2}) x

= 5 x - x (6 - x^{2})

Expand

- x (6 - x^{2}) : - 6 x + x^{3}

- x (6 - x^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = - x, b = 6, c = x^{2}

= - x \cdot 6 - (- x) x^{2}

Apply minus-plus rules

- (- a) = a

= - 6 x + x^{2} x

x^{2} x = x^{3}

x^{2} x

Apply exponent rule:

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

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

= x^{2 + 1}

Add the numbers:

2 + 1 = 3

= x^{3}

= - 6 x + x^{3}

= 5 x - 6 x + x^{3}

Add similar elements:

5 x - 6 x = - x

= - x + x^{3}

= \frac{- x + x ^{3}}{6 - x ^{2}}

= \frac{\frac{- x + x ^{3}}{6 - x ^{2}}}{\frac{6 + 4 x ^{2}}{6 - x ^{2}}}

Divide fractions:

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

= \frac{( - x + x ^{3} ) ( 6 - x ^{2} )}{( 6 - x ^{2} ) ( 6 + 4 x ^{2} )}

Cancel the common factor:

6 - x^{2}

= \frac{- x + x ^{3}}{6 + 4 x ^{2}}

\frac{- x + x ^{3}}{6 + 4 x ^{2}} = 0

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

- x + x^{3} = 0

Solve

- x + x^{3} = 0 : x = 0, x = - 1, x = 1

- x + x^{3} = 0

Factor

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

- x + x^{3}

Factor out common term

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

x^{3} - x

Apply exponent rule:

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

x^{3} = x^{2} x

= x^{2} x - x

Factor out common term

x

= x (x^{2} - 1)

= x (x^{2} - 1)

Factor

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

x^{2} - 1

Rewrite

1

1^{2}

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

Apply Difference of Two Squares Formula:

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

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

= (x + 1) (x - 1)

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

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

x = 0 or x + 1 = 0 or x - 1 = 0

Solve

x + 1 = 0 : x = - 1

x + 1 = 0

Move

1

to the right side

x + 1 = 0

Subtract

1

from both sides

x + 1 - 1 = 0 - 1

Simplify

x = - 1

x = - 1

Solve

x - 1 = 0 : x = 1

x - 1 = 0

Move

1

to the right side

x - 1 = 0

Add

1

to both sides

x - 1 + 1 = 0 + 1

Simplify

x = 1

x = 1

The solutions are

x = 0, x = - 1, x = 1

x = 0, x = - 1, x = 1

Verify Solutions

Find undefined (singularity) points:

x = 6, x = - 6

Take the denominator(s) of

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x}

and compare to zero

Solve

1 - \frac{x}{3} \cdot \frac{x}{2} = 0 : x = 6, x = - 6

1 - \frac{x}{3} \cdot \frac{x}{2} = 0

Move

1

to the right side

1 - \frac{x}{3} \cdot \frac{x}{2} = 0

Subtract

1

from both sides

1 - \frac{x}{3} \cdot \frac{x}{2} - 1 = 0 - 1

Simplify

- \frac{x}{3} \cdot \frac{x}{2} = - 1

- \frac{x}{3} \cdot \frac{x}{2} = - 1

Simplify

- \frac{x ^{2}}{6} = - 1

Multiply both sides by

- 6

(- \frac{x ^{2}}{6}) (- 6) = (- 1) (- 6)

x^{2} = 6

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

x = 6, x = - 6

The following points are undefined

x = 6, x = - 6

Combine undefined points with solutions:

x = 0, x = - 1, x = 1

x = 0, x = - 1, x = 1

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

arctan (\frac{x}{3}) + arctan (\frac{x}{2}) = arctan (x)

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

Check the solution

0 :

True

0

Plug in

n = 1

0

For

arctan (\frac{x}{3}) + arctan (\frac{x}{2}) = arctan (x)

plug in

x = 0

arctan (\frac{0}{3}) + arctan (\frac{0}{2}) = arctan (0)

Refine

0 = 0

\Rightarrow True

Check the solution

- 1 :

True

- 1

Plug in

n = 1

- 1

For

arctan (\frac{x}{3}) + arctan (\frac{x}{2}) = arctan (x)

plug in

x = - 1

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

Refine

- 0.78539 \dots = - 0.78539 \dots

\Rightarrow True

Check the solution

1 :

True

1

Plug in

n = 1

1

For

arctan (\frac{x}{3}) + arctan (\frac{x}{2}) = arctan (x)

plug in

x = 1

arctan (\frac{1}{3}) + arctan (\frac{1}{2}) = arctan (1)

Refine

0.78539 \dots = 0.78539 \dots

\Rightarrow True

x = 0, x = - 1, x = 1

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Frequently Asked Questions (FAQ)

What is the general solution for arctan(x/3)+arctan(x/2)=arctan(x) ?
The general solution for arctan(x/3)+arctan(x/2)=arctan(x) is x=0,x=-1,x=1