Is (1+tan(x))/(1-tan(x))=tan(x+pi/4) ?

The answer to whether (1+tan(x))/(1-tan(x))=tan(x+pi/4) is True

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prove (1+tan(x))/(1-tan(x))=tan(x+pi/4)

Solution

prove

\frac{1 + tan ( x )}{1 - tan ( x )} = tan (x + \frac{π}{4})

Solution

True

Solution steps

\frac{1 + tan ( x )}{1 - tan ( x )} = tan (x + \frac{π}{4})

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

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

Convert element to fraction:

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

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

Multiply:

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

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

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

Join

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

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

Convert element to fraction:

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

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

Multiply:

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

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

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

Divide fractions:

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

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

Cancel the common factor:

cos (x)

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

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

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

Manipulating right side

tan (x + \frac{π}{4})

Rewrite using trig identities

tan (x + \frac{π}{4})

Use the basic trigonometric identity:

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

= \frac{sin ( x + \frac{π}{4} )}{cos ( x + \frac{π}{4} )}

Use the Angle Sum identity:

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

= \frac{sin ( x ) cos ( \frac{π}{4} ) + cos ( x ) sin ( \frac{π}{4} )}{cos ( x + \frac{π}{4} )}

Use the Angle Sum identity:

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

= \frac{sin ( x ) cos ( \frac{π}{4} ) + cos ( x ) sin ( \frac{π}{4} )}{cos ( x ) cos ( \frac{π}{4} ) - sin ( x ) sin ( \frac{π}{4} )}

Simplify

\frac{sin ( x ) cos ( \frac{π}{4} ) + cos ( x ) sin ( \frac{π}{4} )}{cos ( x ) cos ( \frac{π}{4} ) - sin ( x ) sin ( \frac{π}{4} )} : \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )}

\frac{sin ( x ) cos ( \frac{π}{4} ) + cos ( x ) sin ( \frac{π}{4} )}{cos ( x ) cos ( \frac{π}{4} ) - sin ( x ) sin ( \frac{π}{4} )}

sin (x) cos (\frac{π}{4}) + cos (x) sin (\frac{π}{4}) = \frac{2}{2} sin (x) + \frac{2}{2} cos (x)

sin (x) cos (\frac{π}{4}) + cos (x) sin (\frac{π}{4})

Simplify

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{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{2}{2}

= \frac{2}{2} sin (x) + sin (\frac{π}{4}) cos (x)

Simplify

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{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{2}{2}

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

= \frac{\frac{2}{2} sin ( x ) + \frac{2}{2} cos ( x )}{cos ( \frac{π}{4} ) cos ( x ) - sin ( \frac{π}{4} ) sin ( x )}

cos (x) cos (\frac{π}{4}) - sin (x) sin (\frac{π}{4}) = \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

cos (x) cos (\frac{π}{4}) - sin (x) sin (\frac{π}{4})

Simplify

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{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{2}{2}

= \frac{2}{2} cos (x) - sin (\frac{π}{4}) sin (x)

Simplify

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{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{2}{2}

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Combine the fractions

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

Apply rule

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

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

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

Combine the fractions

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

Apply rule

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

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

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

Divide fractions:

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

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

Cancel the common factor:

2

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

Factor out common term

2

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

Factor out common term

2

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

Cancel the common factor:

2

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is (1+tan(x))/(1-tan(x))=tan(x+pi/4) ?
The answer to whether (1+tan(x))/(1-tan(x))=tan(x+pi/4) is True