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شائع علم المثلثات >

prove csc(x)(sin(x)+tan(x))=1+sec(x)

الحلّ

prove

csc (x) (sin (x) + tan (x)) = 1 + sec (x)

الحلّ

صحيح

خطوات الحلّ

csc (x) (sin (x) + tan (x)) = 1 + sec (x)

قم بالعمل على الطرف الأيسر

csc (x) (sin (x) + tan (x))

sin, cos :

عبّر بواسطة

(sin (x) + tan (x)) csc (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

:اضرب كسور

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

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

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

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

اضرب

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

(a) = a

:احذف الأقواس

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

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

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

: استخدم ميزات الكسور التالية

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

sin (x)

قم باخراج العامل المشترك

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

sin (x) :

إلغ العوامل المشتركة

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

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

= \frac{1 + cos ( x )}{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 )}}{\frac{1}{s e c ( x )}}

بسّط

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

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

: استخدم ميزات الكسور التالية

= \frac{( 1 + \frac{1}{s e c ( x )} ) sec ( x )}{1}

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

اضرب

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

= \frac{\frac{s e c ( x ) + 1}{s e c ( x )} sec ( x )}{1}

\frac{a}{1} = a

: استخدم ميزات الكسور التالية

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

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

:اضرب كسور

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

sec (x) :

إلغ العوامل المشتركة

= sec (x) + 1

sec (x) + 1

sec (x) + 1

= 1 + sec (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

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