Is tan(pi/4+θ)tan((3pi)/4+θ)=-1 ?

The answer to whether tan(pi/4+θ)tan((3pi)/4+θ)=-1 is True

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prove tan(pi/4+θ)tan((3pi)/4+θ)=-1

Solution

prove

tan (\frac{π}{4} + θ) tan (\frac{3 π}{4} + θ) = - 1

Solution

True

Solution steps

tan (\frac{π}{4} + θ) tan (\frac{3 π}{4} + θ) = - 1

Manipulating left side

tan (\frac{π}{4} + θ) tan (\frac{3 π}{4} + θ)

Rewrite using trig identities

tan (\frac{π}{4} + θ)

Use the basic trigonometric identity:

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

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

Use the Angle Sum identity:

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

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

Use the Angle Sum identity:

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

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

Simplify

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

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

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

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

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) + \frac{2}{2} sin (θ)

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

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

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

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

Multiply

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

\frac{2}{2} cos (θ)

Multiply fractions:

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

= \frac{2 cos ( θ )}{2}

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

Multiply

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

\frac{2}{2} sin (θ)

Multiply fractions:

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

= \frac{2 sin ( θ )}{2}

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

Multiply

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

\frac{2}{2} cos (θ)

Multiply fractions:

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

= \frac{2 cos ( θ )}{2}

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

Multiply

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

\frac{2}{2} sin (θ)

Multiply fractions:

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

= \frac{2 sin ( θ )}{2}

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

Cancel the common factor:

2

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

Factor out common term

2

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

Factor out common term

2

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

Cancel the common factor:

2

= \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 )} tan (\frac{3 π}{4} + θ)

Rewrite using trig identities

tan (\frac{3 π}{4} + θ)

Use the basic trigonometric identity:

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

= \frac{sin ( \frac{3 π}{4} + θ )}{cos ( \frac{3 π}{4} + θ )}

Use the Angle Sum identity:

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

= \frac{sin ( \frac{3 π}{4} ) cos ( θ ) + cos ( \frac{3 π}{4} ) sin ( θ )}{cos ( \frac{3 π}{4} + θ )}

Use the Angle Sum identity:

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

= \frac{sin ( \frac{3 π}{4} ) cos ( θ ) + cos ( \frac{3 π}{4} ) sin ( θ )}{cos ( \frac{3 π}{4} ) cos ( θ ) - sin ( \frac{3 π}{4} ) sin ( θ )}

Simplify

\frac{sin ( \frac{3 π}{4} ) cos ( θ ) + cos ( \frac{3 π}{4} ) sin ( θ )}{cos ( \frac{3 π}{4} ) cos ( θ ) - sin ( \frac{3 π}{4} ) sin ( θ )} : - \frac{cos ( θ ) - sin ( θ )}{cos ( θ ) + sin ( θ )}

\frac{sin ( \frac{3 π}{4} ) cos ( θ ) + cos ( \frac{3 π}{4} ) sin ( θ )}{cos ( \frac{3 π}{4} ) cos ( θ ) - sin ( \frac{3 π}{4} ) sin ( θ )}

sin (\frac{3 π}{4}) cos (θ) + cos (\frac{3 π}{4}) sin (θ) = \frac{2}{2} cos (θ) - \frac{2}{2} sin (θ)

sin (\frac{3 π}{4}) cos (θ) + cos (\frac{3 π}{4}) sin (θ)

Simplify

sin (\frac{3 π}{4}) : \frac{2}{2}

sin (\frac{3 π}{4})

Use the following trivial identity:

sin (\frac{3 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) + cos (\frac{3 π}{4}) sin (θ)

Simplify

cos (\frac{3 π}{4}) : - \frac{2}{2}

cos (\frac{3 π}{4})

Use the following trivial identity:

cos (\frac{3 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) - \frac{2}{2} sin (θ)

= \frac{\frac{2}{2} cos ( θ ) - \frac{2}{2} sin ( θ )}{cos ( \frac{3 π}{4} ) cos ( θ ) - sin ( \frac{3 π}{4} ) sin ( θ )}

cos (\frac{3 π}{4}) cos (θ) - sin (\frac{3 π}{4}) sin (θ) = - \frac{2}{2} cos (θ) - \frac{2}{2} sin (θ)

cos (\frac{3 π}{4}) cos (θ) - sin (\frac{3 π}{4}) sin (θ)

Simplify

cos (\frac{3 π}{4}) : - \frac{2}{2}

cos (\frac{3 π}{4})

Use the following trivial identity:

cos (\frac{3 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{3 π}{4}) sin (θ)

Simplify

sin (\frac{3 π}{4}) : \frac{2}{2}

sin (\frac{3 π}{4})

Use the following trivial identity:

sin (\frac{3 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} cos ( θ ) - \frac{2}{2} sin ( θ )}{- \frac{2}{2} cos ( θ ) - \frac{2}{2} sin ( θ )}

Multiply

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

\frac{2}{2} cos (θ)

Multiply fractions:

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

= \frac{2 cos ( θ )}{2}

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

Multiply

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

\frac{2}{2} sin (θ)

Multiply fractions:

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

= \frac{2 sin ( θ )}{2}

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

Multiply

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

\frac{2}{2} cos (θ)

Multiply fractions:

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

= \frac{2 cos ( θ )}{2}

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

Multiply

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

\frac{2}{2} sin (θ)

Multiply fractions:

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

= \frac{2 sin ( θ )}{2}

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

Cancel the common factor:

2

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

Factor out common term

2

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

Factor out common term

2

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

Cancel the common factor:

2

= - \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 )} (- \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 )}) = - 1

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

Remove parentheses:

(- a) = - a

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

Multiply fractions:

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

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

Cancel the common factor:

cos (θ) + sin (θ)

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

Cancel the common factor:

cos (θ) - sin (θ)

= - 1

= - 1

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is tan(pi/4+θ)tan((3pi)/4+θ)=-1 ?
The answer to whether tan(pi/4+θ)tan((3pi)/4+θ)=-1 is True