What is the value of tan(arctan(4/3)-arctan(1/7)) ?

The value of tan(arctan(4/3)-arctan(1/7)) is 1

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tan(arctan(4/3)-arctan(1/7))

Solution

tan (arctan (\frac{4}{3}) - arctan (\frac{1}{7}))

Solution

1

Solution steps

tan (arctan (\frac{4}{3}) - arctan (\frac{1}{7}))

Rewrite using trig identities:

arctan (\frac{4}{3}) - arctan (\frac{1}{7}) = arctan (1)

arctan (\frac{4}{3}) - arctan (\frac{1}{7})

Use the Sum to Product identity:

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

= arctan (\frac{\frac{4}{3} - \frac{1}{7}}{1 + \frac{4}{3} \cdot \frac{1}{7}})

Simplify:

\frac{\frac{4}{3} - \frac{1}{7}}{1 + \frac{4}{3} \cdot \frac{1}{7}} = 1

\frac{\frac{4}{3} - \frac{1}{7}}{1 + \frac{4}{3} \cdot \frac{1}{7}}

\frac{4}{3} \cdot \frac{1}{7} = \frac{4}{21}

\frac{4}{3} \cdot \frac{1}{7}

Multiply fractions:

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

= \frac{4 \cdot 1}{3 \cdot 7}

Multiply the numbers:

4 \cdot 1 = 4

= \frac{4}{3 \cdot 7}

Multiply the numbers:

3 \cdot 7 = 21

= \frac{4}{21}

= \frac{\frac{4}{3} - \frac{1}{7}}{1 + \frac{4}{21}}

Join

\frac{4}{3} - \frac{1}{7} : \frac{25}{21}

\frac{4}{3} - \frac{1}{7}

Least Common Multiplier of

3, 7 : 21

3, 7

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

7 : 7

7

7

is a prime number, therefore no factorization is possible

= 7

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

3

7

= 3 \cdot 7

Multiply the numbers:

3 \cdot 7 = 21

= 21

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

21

For

\frac{4}{3} :

multiply the denominator and numerator by

7

\frac{4}{3} = \frac{4 \cdot 7}{3 \cdot 7} = \frac{28}{21}

For

\frac{1}{7} :

multiply the denominator and numerator by

3

\frac{1}{7} = \frac{1 \cdot 3}{7 \cdot 3} = \frac{3}{21}

= \frac{28}{21} - \frac{3}{21}

Since the denominators are equal, combine the fractions:

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

= \frac{28 - 3}{21}

Subtract the numbers:

28 - 3 = 25

= \frac{25}{21}

= \frac{\frac{25}{21}}{1 + \frac{4}{21}}

Apply the fraction rule:

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

= \frac{25}{21 ( 1 + \frac{4}{21} )}

Join

1 + \frac{4}{21} : \frac{25}{21}

1 + \frac{4}{21}

Convert element to fraction:

1 = \frac{1 \cdot 21}{21}

= \frac{1 \cdot 21}{21} + \frac{4}{21}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 21 + 4}{21}

1 \cdot 21 + 4 = 25

1 \cdot 21 + 4

Multiply the numbers:

1 \cdot 21 = 21

= 21 + 4

Add the numbers:

21 + 4 = 25

= 25

= \frac{25}{21}

= \frac{25}{21 \cdot \frac{25}{21}}

Multiply

21 \cdot \frac{25}{21} : 25

21 \cdot \frac{25}{21}

Multiply fractions:

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

= \frac{25 \cdot 21}{21}

Cancel the common factor:

21

= 25

= \frac{25}{25}

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(4/3)-arctan(1/7)) ?
The value of tan(arctan(4/3)-arctan(1/7)) is 1