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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Use the negative angle identity:

sin (- x) = - sin (x)

= \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 )}

Apply rule

- (- a) = a

= \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

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