What is the value of tan(arctan(1/4)+arctan(3/5)) ?

The value of tan(arctan(1/4)+arctan(3/5)) is 1

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tan(arctan(1/4)+arctan(3/5))

Solution

tan (arctan (\frac{1}{4}) + arctan (\frac{3}{5}))

Solution

1

Solution steps

tan (arctan (\frac{1}{4}) + arctan (\frac{3}{5}))

Rewrite using trig identities:

arctan (\frac{1}{4}) + arctan (\frac{3}{5}) = arctan (1)

arctan (\frac{1}{4}) + arctan (\frac{3}{5})

Use the Sum to Product identity:

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

= arctan (\frac{\frac{1}{4} + \frac{3}{5}}{1 - \frac{1}{4} \cdot \frac{3}{5}})

Simplify:

\frac{\frac{1}{4} + \frac{3}{5}}{1 - \frac{1}{4} \cdot \frac{3}{5}} = 1

\frac{\frac{1}{4} + \frac{3}{5}}{1 - \frac{1}{4} \cdot \frac{3}{5}}

\frac{1}{4} \cdot \frac{3}{5} = \frac{3}{20}

\frac{1}{4} \cdot \frac{3}{5}

Multiply fractions:

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

= \frac{1 \cdot 3}{4 \cdot 5}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{4 \cdot 5}

Multiply the numbers:

4 \cdot 5 = 20

= \frac{3}{20}

= \frac{\frac{1}{4} + \frac{3}{5}}{1 - \frac{3}{20}}

Join

\frac{1}{4} + \frac{3}{5} : \frac{17}{20}

\frac{1}{4} + \frac{3}{5}

Least Common Multiplier of

4, 5 : 20

4, 5

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

5 : 5

5

5

is a prime number, therefore no factorization is possible

= 5

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

4

5

= 2 \cdot 2 \cdot 5

Multiply the numbers:

2 \cdot 2 \cdot 5 = 20

= 20

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

20

For

\frac{1}{4} :

multiply the denominator and numerator by

5

\frac{1}{4} = \frac{1 \cdot 5}{4 \cdot 5} = \frac{5}{20}

For

\frac{3}{5} :

multiply the denominator and numerator by

4

\frac{3}{5} = \frac{3 \cdot 4}{5 \cdot 4} = \frac{12}{20}

= \frac{5}{20} + \frac{12}{20}

Since the denominators are equal, combine the fractions:

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

= \frac{5 + 12}{20}

Add the numbers:

5 + 12 = 17

= \frac{17}{20}

= \frac{\frac{17}{20}}{1 - \frac{3}{20}}

Apply the fraction rule:

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

= \frac{17}{20 ( 1 - \frac{3}{20} )}

Join

1 - \frac{3}{20} : \frac{17}{20}

1 - \frac{3}{20}

Convert element to fraction:

1 = \frac{1 \cdot 20}{20}

= \frac{1 \cdot 20}{20} - \frac{3}{20}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 20 - 3}{20}

1 \cdot 20 - 3 = 17

1 \cdot 20 - 3

Multiply the numbers:

1 \cdot 20 = 20

= 20 - 3

Subtract the numbers:

20 - 3 = 17

= 17

= \frac{17}{20}

= \frac{17}{20 \cdot \frac{17}{20}}

Multiply

20 \cdot \frac{17}{20} : 17

20 \cdot \frac{17}{20}

Multiply fractions:

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

= \frac{17 \cdot 20}{20}

Cancel the common factor:

20

= 17

= \frac{17}{17}

Apply rule

\frac{a}{a} = 1

= 1

= arctan (1)

= tan (arctan (1))

Rewrite using trig identities:

tan (arctan (1)) = 1

Use the following identity:

tan (arctan (x)) = x

= 1

= 1

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Frequently Asked Questions (FAQ)

What is the value of tan(arctan(1/4)+arctan(3/5)) ?
The value of tan(arctan(1/4)+arctan(3/5)) is 1