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

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

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

tan (θ + \frac{π}{4})

Rewrite using trig identities

tan (θ + \frac{π}{4})

Use the basic trigonometric identity:

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

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

Use the Angle Sum identity:

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

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

Use the Angle Sum identity:

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

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

Simplify

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

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

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

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

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 (θ) + \frac{2}{2} cos (θ)

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

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

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

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 (θ) - \frac{2}{2} sin (θ)

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

Multiply

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

cos (θ) \frac{2}{2}

Multiply fractions:

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

= \frac{2 cos ( θ )}{2}

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

Multiply

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

sin (θ) \frac{2}{2}

Multiply fractions:

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

= \frac{2 sin ( θ )}{2}

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

Multiply

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

sin (θ) \frac{2}{2}

Multiply fractions:

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

= \frac{2 sin ( θ )}{2}

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

Multiply

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

cos (θ) \frac{2}{2}

Multiply fractions:

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

= \frac{2 cos ( θ )}{2}

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

Combine the fractions

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

Apply rule

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

= \frac{2 cos ( θ ) - 2 sin ( θ )}{2}

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

Combine the fractions

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

Apply rule

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

= \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

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

Divide fractions:

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

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

Cancel the common factor:

2

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

Factor out common term

2

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

Factor out common term

2

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

Cancel the common factor:

2

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

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

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

Manipulating right side

\frac{1 + tan ( θ )}{1 - tan ( θ )}

Express with sin, cos

\frac{1 + tan ( θ )}{1 - tan ( θ )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

1 - \frac{sin ( θ )}{cos ( θ )}

Convert element to fraction:

1 = \frac{1 cos ( θ )}{cos ( θ )}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (θ) = cos (θ)

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

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

Join

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

1 + \frac{sin ( θ )}{cos ( θ )}

Convert element to fraction:

1 = \frac{1 cos ( θ )}{cos ( θ )}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (θ) = cos (θ)

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

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

Divide fractions:

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

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

Cancel the common factor:

cos (θ)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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