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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

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

Factor out common term

sin (x)

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

Cancel the common factor:

cos (x) + 1

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

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

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

Use the basic trigonometric identity:

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

= tan (x)

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

1 + cos^{2} (θ) = 2 cos^{2} (θ) + sin^{2} (θ)

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

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