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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

tan (\frac{π}{4} - t)

Rewrite using trig identities

tan (\frac{π}{4} - t)

Use the basic trigonometric identity:

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

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

Use the Angle Difference identity:

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

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

Use the Angle Difference identity:

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

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

Simplify

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

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

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

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

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 (t) - cos (\frac{π}{4}) sin (t)

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 (t) - \frac{2}{2} sin (t)

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

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

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

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 (t) + sin (\frac{π}{4}) sin (t)

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 (t) + \frac{2}{2} sin (t)

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

Multiply

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

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

Multiply fractions:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Combine the fractions

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

Apply rule

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

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

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

Combine the fractions

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

Apply rule

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

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

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

Divide fractions:

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

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

Cancel the common factor:

2

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

Factor out common term

2

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

Factor out common term

2

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

Cancel the common factor:

2

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

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

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

Manipulating right side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (t) = cos (t)

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

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

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (t) = cos (t)

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

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

Divide fractions:

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

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

Cancel the common factor:

cos (t)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

Popular Examples

prove-tan(x)=tan(-x)prove

- tan (x) = tan (- x)

prove (csc(θ)+sin(θ))/(tan(θ)+1)=csc(θ)prove

\frac{csc ( θ ) + sin ( θ )}{tan ( θ ) + 1} = csc (θ)

prove-sin^2(x)-cos^2(x)=-1 prove

- sin^{2} (x) - cos^{2} (x) = - 1

prove sin(3θ)=3cos^2(θ)sin(θ)-sin^3(θ)prove

sin (3 θ) = 3 cos^{2} (θ) sin (θ) - sin^{3} (θ)

prove (tan(θ)cos(θ))/(sin(θ))=1 prove

\frac{tan ( θ ) cos ( θ )}{sin ( θ )} = 1

Frequently Asked Questions (FAQ)

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