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العربية

prove (csc(x)+1)/(cot(x)+cos(x))=sec(x)

الحلّ

prove

\frac{csc ( x ) + 1}{cot ( x ) + cos ( x )} = sec (x)

صحيح

خطوات الحلّ

\frac{csc ( x ) + 1}{cot ( x ) + cos ( x )} = sec (x)

قم بالعمل على الطرف الأيسر

\frac{csc ( x ) + 1}{cot ( x ) + cos ( x )}

sin, cos :

عبّر بواسطة

\frac{1 + csc ( x )}{cos ( x ) + cot ( x )}

csc (x) = \frac{1}{sin ( x )}

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

\frac{1}{cos ( x )}

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

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

1 \cdot sin (x) = sin (x) :

اضرب

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

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

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

:اقسم الكسور

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

sin (x) :

إلغ العوامل المشتركة

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

cos (x)

قم باخراج العامل المشترك

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

sin (x) + 1 :

إلغ العوامل المشتركة

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

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

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

Rewrite using trig identities

cos (x) = \frac{1}{sec ( x )}

:Use the basic trigonometric identity

\frac{1}{\frac{1}{s e c ( x )}}

بسّط

\frac{1}{\frac{1}{s e c ( x )}}

\frac{1}{\frac{b}{c}} = \frac{c}{b}

: استخدم ميزات الكسور التالية

= \frac{sec ( x )}{1}

\frac{a}{1} = a

فعّل القانون

= sec (x)

sec (x)

sec (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e csc^{2} (\frac{θ}{2}) = \frac{2}{1 - cos ( θ )}

p ro v e sec (x) cos (x) + tan^{2} (x) = sec^{2} (x)

p ro v e cot^{2} (θ) + 1 = \frac{1}{sin ^{2} ( θ )}

p ro v e 3 cos (2 x + \frac{π}{2}) - 2 = 6 cos^{2} (x + \frac{π}{4})

p ro v e cos^{4} (b) - sin^{4} (b) = cos (2 b)