Is 1/(cot(x)+1)=(tan(x))/(tan(x)+1) ?

The answer to whether 1/(cot(x)+1)=(tan(x))/(tan(x)+1) is True

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prove 1/(cot(x)+1)=(tan(x))/(tan(x)+1)

Solution

prove

\frac{1}{cot ( x ) + 1} = \frac{tan ( x )}{tan ( x ) + 1}

Solution

True

Solution steps

\frac{1}{cot ( x ) + 1} = \frac{tan ( x )}{tan ( x ) + 1}

Manipulating left side

\frac{1}{cot ( x ) + 1}

Express with sin, cos

\frac{1}{1 + cot ( x )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

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

Convert element to fraction:

1 = \frac{1 sin ( x )}{sin ( x )}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot sin (x) = sin (x)

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

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

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

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

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

Manipulating right side

\frac{tan ( x )}{tan ( x ) + 1}

Express with sin, cos

\frac{tan ( x )}{1 + tan ( x )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

= \frac{sin ( x )}{cos ( x ) ( 1 + \frac{s i n ( x )}{c o s ( x )} )}

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (x) = cos (x)

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

= \frac{sin ( x )}{\frac{c o s ( x ) + s i n ( x )}{c o s ( x )} cos ( x )}

Multiply

cos (x) \frac{cos ( x ) + sin ( x )}{cos ( x )} : cos (x) + sin (x)

cos (x) \frac{cos ( x ) + sin ( x )}{cos ( x )}

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= cos (x) + sin (x)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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