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العربية

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

الحلّ

prove

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

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

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

sin (- x) = - sin (x)

:Use the negative angle identity

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

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

- (- a) = a

فعّل القانون

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

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \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 )}

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

: استخدم ميزات الكسور التالية

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

sin (x)

قم باخراج العامل المشترك

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

cos (x) + 1 :

إلغ العوامل المشتركة

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

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

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

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

:Use the basic trigonometric identity

= tan (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e tan (2 a) = \frac{2 tan ( a )}{1 - tan ^{2} ( a )}

p ro v e cos (π - θ) + sin (\frac{π}{2} + θ) = 0

p ro v e tan (y) + cot (y) = sec (y) csc (y)

p ro v e \frac{1}{sin ( x )} = csc (x)

p ro v e 2 csc (2 x) = csc^{2} (x) tan (x)