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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

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

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

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

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