Is tan(x+pi/3)=(tan(x)+sqrt(3))/(1-sqrt(3)tan(x)) ?

The answer to whether tan(x+pi/3)=(tan(x)+sqrt(3))/(1-sqrt(3)tan(x)) is True

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prove tan(x+pi/3)=(tan(x)+sqrt(3))/(1-sqrt(3)tan(x))

Solution

prove

tan (x + \frac{π}{3}) = \frac{tan ( x ) + 3}{1 - 3 tan ( x )}

Solution

True

Solution steps

tan (x + \frac{π}{3}) = \frac{tan ( x ) + 3}{1 - 3 tan ( x )}

Manipulating left side

tan (x + \frac{π}{3})

Rewrite using trig identities

tan (x + \frac{π}{3})

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{sin ( x + \frac{π}{3} )}{cos ( x + \frac{π}{3} )}

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= \frac{sin ( x ) cos ( \frac{π}{3} ) + cos ( x ) sin ( \frac{π}{3} )}{cos ( x + \frac{π}{3} )}

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= \frac{sin ( x ) cos ( \frac{π}{3} ) + cos ( x ) sin ( \frac{π}{3} )}{cos ( x ) cos ( \frac{π}{3} ) - sin ( x ) sin ( \frac{π}{3} )}

Simplify

\frac{sin ( x ) cos ( \frac{π}{3} ) + cos ( x ) sin ( \frac{π}{3} )}{cos ( x ) cos ( \frac{π}{3} ) - sin ( x ) sin ( \frac{π}{3} )} : \frac{\frac{1}{2} sin ( x ) + \frac{3}{2} cos ( x )}{\frac{1}{2} cos ( x ) - \frac{3}{2} sin ( x )}

\frac{sin ( x ) cos ( \frac{π}{3} ) + cos ( x ) sin ( \frac{π}{3} )}{cos ( x ) cos ( \frac{π}{3} ) - sin ( x ) sin ( \frac{π}{3} )}

sin (x) cos (\frac{π}{3}) + cos (x) sin (\frac{π}{3}) = \frac{1}{2} sin (x) + \frac{3}{2} cos (x)

sin (x) cos (\frac{π}{3}) + cos (x) sin (\frac{π}{3})

Simplify

cos (\frac{π}{3}) : \frac{1}{2}

cos (\frac{π}{3})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (x)

periodicity table with

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

= \frac{1}{2} sin (x) + sin (\frac{π}{3}) cos (x)

Simplify

sin (\frac{π}{3}) : \frac{3}{2}

sin (\frac{π}{3})

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= \frac{1}{2} sin (x) + \frac{3}{2} cos (x)

= \frac{\frac{1}{2} sin ( x ) + \frac{3}{2} cos ( x )}{cos ( \frac{π}{3} ) cos ( x ) - sin ( \frac{π}{3} ) sin ( x )}

cos (x) cos (\frac{π}{3}) - sin (x) sin (\frac{π}{3}) = \frac{1}{2} cos (x) - \frac{3}{2} sin (x)

cos (x) cos (\frac{π}{3}) - sin (x) sin (\frac{π}{3})

Simplify

cos (\frac{π}{3}) : \frac{1}{2}

cos (\frac{π}{3})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (x)

periodicity table with

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

= \frac{1}{2} cos (x) - sin (\frac{π}{3}) sin (x)

Simplify

sin (\frac{π}{3}) : \frac{3}{2}

sin (\frac{π}{3})

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= \frac{1}{2} cos (x) - \frac{3}{2} sin (x)

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

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

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

Simplify

\frac{cos ( x ) \frac{3}{2} + \frac{1}{2} sin ( x )}{cos ( x ) \frac{1}{2} - \frac{3}{2} sin ( x )} : \frac{3 cos ( x ) + sin ( x )}{cos ( x ) - 3 sin ( x )}

\frac{cos ( x ) \frac{3}{2} + \frac{1}{2} sin ( x )}{cos ( x ) \frac{1}{2} - \frac{3}{2} sin ( x )}

Multiply

cos (x) \frac{1}{2} : \frac{cos ( x )}{2}

cos (x) \frac{1}{2}

Multiply fractions:

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

= \frac{cos ( x )}{2}

= \frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{c o s ( x )}{2} - \frac{3}{2} sin ( x )}

Multiply

\frac{3}{2} sin (x) : \frac{3 sin ( x )}{2}

\frac{3}{2} sin (x)

Multiply fractions:

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

= \frac{3 sin ( x )}{2}

= \frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{c o s ( x )}{2} - \frac{3 s i n ( x )}{2}}

Multiply

cos (x) \frac{3}{2} : \frac{3 cos ( x )}{2}

cos (x) \frac{3}{2}

Multiply fractions:

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

= \frac{3 cos ( x )}{2}

= \frac{\frac{3 c o s ( x )}{2} + \frac{1}{2} sin ( x )}{\frac{c o s ( x )}{2} - \frac{3 s i n ( x )}{2}}

Multiply

\frac{1}{2} sin (x) : \frac{sin ( x )}{2}

\frac{1}{2} sin (x)

Multiply fractions:

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

= \frac{sin ( x )}{2}

= \frac{\frac{3 c o s ( x )}{2} + \frac{s i n ( x )}{2}}{\frac{c o s ( x )}{2} - \frac{3 s i n ( x )}{2}}

Combine the fractions

\frac{cos ( x )}{2} - \frac{3 sin ( x )}{2} : \frac{cos ( x ) - 3 sin ( x )}{2}

Apply rule

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

= \frac{cos ( x ) - 3 sin ( x )}{2}

= \frac{\frac{3 c o s ( x )}{2} + \frac{s i n ( x )}{2}}{\frac{c o s ( x ) - 3 s i n ( x )}{2}}

Combine the fractions

\frac{3 cos ( x )}{2} + \frac{sin ( x )}{2} : \frac{3 cos ( x ) + sin ( x )}{2}

Apply rule

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

= \frac{3 cos ( x ) + sin ( x )}{2}

= \frac{\frac{3 c o s ( x ) + s i n ( x )}{2}}{\frac{c o s ( x ) - 3 s i n ( x )}{2}}

Divide fractions:

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

= \frac{( 3 cos ( x ) + sin ( x ) ) \cdot 2}{2 ( cos ( x ) - 3 sin ( x ) )}

Cancel the common factor:

2

= \frac{3 cos ( x ) + sin ( x )}{cos ( x ) - 3 sin ( x )}

= \frac{sin ( x ) + cos ( x ) 3}{cos ( x ) - sin ( x ) 3}

Manipulating right side

\frac{tan ( x ) + 3}{1 - 3 tan ( x )}

Express with sin, cos

\frac{3 + tan ( x )}{1 - 3 tan ( x )}

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{3 + \frac{s i n ( x )}{c o s ( x )}}{1 - 3 \frac{s i n ( x )}{c o s ( x )}}

Simplify

\frac{3 + \frac{s i n ( x )}{c o s ( x )}}{1 - 3 \frac{s i n ( x )}{c o s ( x )}} : \frac{3 cos ( x ) + sin ( x )}{cos ( x ) - 3 sin ( x )}

\frac{3 + \frac{s i n ( x )}{c o s ( x )}}{1 - 3 \frac{s i n ( x )}{c o s ( x )}}

Multiply

3 \frac{sin ( x )}{cos ( x )} : \frac{3 sin ( x )}{cos ( x )}

3 \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

= \frac{sin ( x ) 3}{cos ( x )}

= \frac{3 + \frac{s i n ( x )}{c o s ( x )}}{1 - \frac{3 s i n ( x )}{c o s ( x )}}

Join

3 + \frac{sin ( x )}{cos ( x )} : \frac{3 cos ( x ) + sin ( x )}{cos ( x )}

3 + \frac{sin ( x )}{cos ( x )}

Convert element to fraction:

3 = \frac{3 cos ( x )}{cos ( x )}

= \frac{3 cos ( x )}{cos ( x )} + \frac{sin ( x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{3 cos ( x ) + sin ( x )}{cos ( x )}

= \frac{\frac{3 c o s ( x ) + s i n ( x )}{c o s ( x )}}{1 - \frac{3 s i n ( x )}{c o s ( x )}}

Apply the fraction rule:

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

= \frac{3 cos ( x ) + sin ( x )}{cos ( x ) ( 1 - \frac{s i n ( x ) 3}{c o s ( x )} )}

Join

1 - \frac{sin ( x ) 3}{cos ( x )} : \frac{cos ( x ) - 3 sin ( x )}{cos ( x )}

1 - \frac{sin ( x ) 3}{cos ( x )}

Convert element to fraction:

1 = \frac{1 cos ( x )}{cos ( x )}

= \frac{1 \cdot cos ( x )}{cos ( x )} - \frac{sin ( x ) 3}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot cos ( x ) - sin ( x ) 3}{cos ( x )}

Multiply:

1 \cdot cos (x) = cos (x)

= \frac{cos ( x ) - 3 sin ( x )}{cos ( x )}

= \frac{3 cos ( x ) + sin ( x )}{\frac{c o s ( x ) - 3 s i n ( x )}{c o s ( x )} cos ( x )}

Multiply

cos (x) \frac{cos ( x ) - sin ( x ) 3}{cos ( x )} : cos (x) - 3 sin (x)

cos (x) \frac{cos ( x ) - sin ( x ) 3}{cos ( x )}

Multiply fractions:

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

= \frac{( cos ( x ) - sin ( x ) 3 ) cos ( x )}{cos ( x )}

Cancel the common factor:

cos (x)

= cos (x) - sin (x) 3

= \frac{3 cos ( x ) + sin ( x )}{cos ( x ) - 3 sin ( x )}

= \frac{3 cos ( x ) + sin ( x )}{cos ( x ) - 3 sin ( x )}

= \frac{sin ( x ) + cos ( x ) 3}{cos ( x ) - sin ( x ) 3}

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is tan(x+pi/3)=(tan(x)+sqrt(3))/(1-sqrt(3)tan(x)) ?
The answer to whether tan(x+pi/3)=(tan(x)+sqrt(3))/(1-sqrt(3)tan(x)) is True