Is (cot(4u)-1)/(cot(4u)+1)=(1-tan(4u))/(1+tan(4u)) ?

The answer to whether (cot(4u)-1)/(cot(4u)+1)=(1-tan(4u))/(1+tan(4u)) is True

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prove (cot(4u)-1)/(cot(4u)+1)=(1-tan(4u))/(1+tan(4u))

Solution

prove

\frac{cot ( 4 u ) - 1}{cot ( 4 u ) + 1} = \frac{1 - tan ( 4 u )}{1 + tan ( 4 u )}

Solution

True

Solution steps

\frac{cot ( 4 u ) - 1}{cot ( 4 u ) + 1} = \frac{1 - tan ( 4 u )}{1 + tan ( 4 u )}

Manipulating left side

\frac{cot ( 4 u ) - 1}{cot ( 4 u ) + 1}

Express with sin, cos

\frac{- 1 + cot ( 4 u )}{1 + cot ( 4 u )}

Use the basic trigonometric identity:

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

= \frac{- 1 + \frac{c o s ( 4 u )}{s i n ( 4 u )}}{1 + \frac{c o s ( 4 u )}{s i n ( 4 u )}}

Simplify

\frac{- 1 + \frac{c o s ( 4 u )}{s i n ( 4 u )}}{1 + \frac{c o s ( 4 u )}{s i n ( 4 u )}} : \frac{- sin ( 4 u ) + cos ( 4 u )}{sin ( 4 u ) + cos ( 4 u )}

\frac{- 1 + \frac{c o s ( 4 u )}{s i n ( 4 u )}}{1 + \frac{c o s ( 4 u )}{s i n ( 4 u )}}

Join

1 + \frac{cos ( 4 u )}{sin ( 4 u )} : \frac{sin ( 4 u ) + cos ( 4 u )}{sin ( 4 u )}

1 + \frac{cos ( 4 u )}{sin ( 4 u )}

Convert element to fraction:

1 = \frac{1 sin ( 4 u )}{sin ( 4 u )}

= \frac{1 \cdot sin ( 4 u )}{sin ( 4 u )} + \frac{cos ( 4 u )}{sin ( 4 u )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot sin ( 4 u ) + cos ( 4 u )}{sin ( 4 u )}

Multiply:

1 \cdot sin (4 u) = sin (4 u)

= \frac{sin ( 4 u ) + cos ( 4 u )}{sin ( 4 u )}

= \frac{- 1 + \frac{c o s ( 4 u )}{s i n ( 4 u )}}{\frac{s i n ( 4 u ) + c o s ( 4 u )}{s i n ( 4 u )}}

Join

- 1 + \frac{cos ( 4 u )}{sin ( 4 u )} : \frac{- sin ( 4 u ) + cos ( 4 u )}{sin ( 4 u )}

- 1 + \frac{cos ( 4 u )}{sin ( 4 u )}

Convert element to fraction:

1 = \frac{1 sin ( 4 u )}{sin ( 4 u )}

= - \frac{1 \cdot sin ( 4 u )}{sin ( 4 u )} + \frac{cos ( 4 u )}{sin ( 4 u )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot sin ( 4 u ) + cos ( 4 u )}{sin ( 4 u )}

Multiply:

1 \cdot sin (4 u) = sin (4 u)

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

= \frac{\frac{- s i n ( 4 u ) + c o s ( 4 u )}{s i n ( 4 u )}}{\frac{s i n ( 4 u ) + c o s ( 4 u )}{s i n ( 4 u )}}

Divide fractions:

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

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

Cancel the common factor:

sin (4 u)

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

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

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

Manipulating right side

\frac{1 - tan ( 4 u )}{1 + tan ( 4 u )}

Express with sin, cos

\frac{1 - tan ( 4 u )}{1 + tan ( 4 u )}

Use the basic trigonometric identity:

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

= \frac{1 - \frac{s i n ( 4 u )}{c o s ( 4 u )}}{1 + \frac{s i n ( 4 u )}{c o s ( 4 u )}}

Simplify

\frac{1 - \frac{s i n ( 4 u )}{c o s ( 4 u )}}{1 + \frac{s i n ( 4 u )}{c o s ( 4 u )}} : \frac{cos ( 4 u ) - sin ( 4 u )}{cos ( 4 u ) + sin ( 4 u )}

\frac{1 - \frac{s i n ( 4 u )}{c o s ( 4 u )}}{1 + \frac{s i n ( 4 u )}{c o s ( 4 u )}}

Join

1 + \frac{sin ( 4 u )}{cos ( 4 u )} : \frac{cos ( 4 u ) + sin ( 4 u )}{cos ( 4 u )}

1 + \frac{sin ( 4 u )}{cos ( 4 u )}

Convert element to fraction:

1 = \frac{1 cos ( 4 u )}{cos ( 4 u )}

= \frac{1 \cdot cos ( 4 u )}{cos ( 4 u )} + \frac{sin ( 4 u )}{cos ( 4 u )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot cos ( 4 u ) + sin ( 4 u )}{cos ( 4 u )}

Multiply:

1 \cdot cos (4 u) = cos (4 u)

= \frac{cos ( 4 u ) + sin ( 4 u )}{cos ( 4 u )}

= \frac{1 - \frac{s i n ( 4 u )}{c o s ( 4 u )}}{\frac{c o s ( 4 u ) + s i n ( 4 u )}{c o s ( 4 u )}}

Join

1 - \frac{sin ( 4 u )}{cos ( 4 u )} : \frac{cos ( 4 u ) - sin ( 4 u )}{cos ( 4 u )}

1 - \frac{sin ( 4 u )}{cos ( 4 u )}

Convert element to fraction:

1 = \frac{1 cos ( 4 u )}{cos ( 4 u )}

= \frac{1 \cdot cos ( 4 u )}{cos ( 4 u )} - \frac{sin ( 4 u )}{cos ( 4 u )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot cos ( 4 u ) - sin ( 4 u )}{cos ( 4 u )}

Multiply:

1 \cdot cos (4 u) = cos (4 u)

= \frac{cos ( 4 u ) - sin ( 4 u )}{cos ( 4 u )}

= \frac{\frac{c o s ( 4 u ) - s i n ( 4 u )}{c o s ( 4 u )}}{\frac{c o s ( 4 u ) + s i n ( 4 u )}{c o s ( 4 u )}}

Divide fractions:

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

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

Cancel the common factor:

cos (4 u)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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