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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Join

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

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

Convert element to fraction:

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

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

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

Apply the fraction rule:

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

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

Factor out common term

sin (x)

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

Cancel the common factor:

1 + cos (x)

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

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

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

Use the basic trigonometric identity:

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

= cot (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (3 x) + sin (x) = 2 sin (2 x) cos (x)

cos (A + B) = cos (A) cos (B) - sin (A) sin (B)

tan^{2} (A) + 1 = \frac{sec ( A )}{cos ( A )}

\frac{2 cot ( x )}{1 + cot ^{2} ( x )} = sin (2 x)

tan^{2} (x) - sin^{2} (x) = tan (x) sin (x)

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