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العربية

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

الحلّ

prove

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

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيمن

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

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

sin (x) + cos (x)

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

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

: استخدم ميزات الكسور التالية

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

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

اضرب

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

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

\frac{a}{1} = a

: استخدم ميزات الكسور التالية

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

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

:اضرب كسور

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

sin (x) :

إلغ العوامل المشتركة

= sin (x) + cos (x)

= sin (x) + cos (x)

= cos (x) + sin (x)

= sin (x) + cos (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e sin^{2} (t) = 2 sin (t) cos (t)

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

p ro v e \frac{sin ( 2 x )}{sin ( x )} = \frac{2}{sec ( x )}

p ro v e \frac{2}{1 + cos ( 2 x )} = sec^{2} (x)

p ro v e cos (2 x) = - 2 sin (x^{2}) + 1