Is tan(135+a)=(tan(a)-1)/(tan(a)+1) ?

The answer to whether tan(135+a)=(tan(a)-1)/(tan(a)+1) is True

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prove tan(135+a)=(tan(a)-1)/(tan(a)+1)

Solution

prove

tan (13 5^{\circ} + a) = \frac{tan ( a ) - 1}{tan ( a ) + 1}

Solution

True

Solution steps

tan (13 5^{\circ} + a) = \frac{tan ( a ) - 1}{tan ( a ) + 1}

Manipulating left side

tan (13 5^{\circ} + a)

Rewrite using trig identities

tan (13 5^{\circ} + a)

Use the basic trigonometric identity:

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

= \frac{sin ( 13 5 ^{\circ} + a )}{cos ( 13 5 ^{\circ} + a )}

Use the Angle Sum identity:

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

= \frac{sin ( 13 5 ^{\circ} ) cos ( a ) + cos ( 13 5 ^{\circ} ) sin ( a )}{cos ( 13 5 ^{\circ} + a )}

Use the Angle Sum identity:

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

= \frac{sin ( 13 5 ^{\circ} ) cos ( a ) + cos ( 13 5 ^{\circ} ) sin ( a )}{cos ( 13 5 ^{\circ} ) cos ( a ) - sin ( 13 5 ^{\circ} ) sin ( a )}

Simplify

\frac{sin ( 13 5 ^{\circ} ) cos ( a ) + cos ( 13 5 ^{\circ} ) sin ( a )}{cos ( 13 5 ^{\circ} ) cos ( a ) - sin ( 13 5 ^{\circ} ) sin ( a )} : - \frac{cos ( a ) - sin ( a )}{cos ( a ) + sin ( a )}

\frac{sin ( 13 5 ^{\circ} ) cos ( a ) + cos ( 13 5 ^{\circ} ) sin ( a )}{cos ( 13 5 ^{\circ} ) cos ( a ) - sin ( 13 5 ^{\circ} ) sin ( a )}

sin (13 5^{\circ}) cos (a) + cos (13 5^{\circ}) sin (a) = \frac{2}{2} cos (a) - \frac{2}{2} sin (a)

sin (13 5^{\circ}) cos (a) + cos (13 5^{\circ}) sin (a)

Simplify

sin (13 5^{\circ}) : \frac{2}{2}

sin (13 5^{\circ})

Use the following trivial identity:

sin (13 5^{\circ}) = \frac{2}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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 (a) + cos (13 5^{\circ}) sin (a)

Simplify

cos (13 5^{\circ}) : - \frac{2}{2}

cos (13 5^{\circ})

Use the following trivial identity:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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 (a) - \frac{2}{2} sin (a)

= \frac{\frac{2}{2} cos ( a ) - \frac{2}{2} sin ( a )}{cos ( 13 5 ^{\circ} ) cos ( a ) - sin ( 13 5 ^{\circ} ) sin ( a )}

cos (13 5^{\circ}) cos (a) - sin (13 5^{\circ}) sin (a) = - \frac{2}{2} cos (a) - \frac{2}{2} sin (a)

cos (13 5^{\circ}) cos (a) - sin (13 5^{\circ}) sin (a)

Simplify

cos (13 5^{\circ}) : - \frac{2}{2}

cos (13 5^{\circ})

Use the following trivial identity:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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 (a) - sin (13 5^{\circ}) sin (a)

Simplify

sin (13 5^{\circ}) : \frac{2}{2}

sin (13 5^{\circ})

Use the following trivial identity:

sin (13 5^{\circ}) = \frac{2}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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 (a) - \frac{2}{2} sin (a)

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

Multiply

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

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

Multiply fractions:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Combine the fractions

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

Apply rule

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

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

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

Combine the fractions

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

Apply rule

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

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

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

Divide fractions:

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

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

Cancel the common factor:

2

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

Factor out common term

2

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

Factor out common term

2

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

Cancel the common factor:

2

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

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

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

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

Simplify

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

Manipulating right side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (a) = cos (a)

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

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

Join

- 1 + \frac{sin ( a )}{cos ( a )} : \frac{- cos ( a ) + sin ( a )}{cos ( a )}

- 1 + \frac{sin ( a )}{cos ( a )}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (a) = cos (a)

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

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

Divide fractions:

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

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

Cancel the common factor:

cos (a)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is tan(135+a)=(tan(a)-1)/(tan(a)+1) ?
The answer to whether tan(135+a)=(tan(a)-1)/(tan(a)+1) is True