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العربية

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

الحلّ

prove

\frac{1 + sec ( x )}{csc ( x )} = sin (x) + tan (x)

صحيح

خطوات الحلّ

\frac{1 + sec ( x )}{csc ( x )} = sin (x) + tan (x)

قم بالعمل على الطرف الأيمن

sin (x) + tan (x)

sin, cos :

عبّر بواسطة

sin (x) + tan (x)

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

حلل إلى عوامل:

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

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

sin (x) + cos (x) sin (x)

حلل إلى عوامل:

sin (x) (1 + cos (x))

sin (x) + cos (x) sin (x)

sin (x)

قم باخراج العامل المشترك

= sin (x) (1 + cos (x))

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

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

Rewrite using trig identities

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

بسّط

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

\frac{a}{1} = a

: استخدم ميزات الكسور التالية

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

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

:اضرب كسور

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

sec (x) :

إلغ العوامل المشتركة

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

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

(sec (x) + 1) \cdot 1

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

اضرب

= (sec (x) + 1)

(a) = a

:احذف الأقواس

= sec (x) + 1

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

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

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

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

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e (cot (x)) (sec (x)) (sin (x)) = 1

p ro v e \frac{cos ( a )}{1 - sin ( a )} - tan (a) = sec (a)

p ro v e tan (\frac{θ}{2}) + cot (\frac{θ}{2}) = 2 csc (θ)

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

p ro v e \frac{cos ( 4 x ) + cos ( 6 x )}{sin ( 4 x ) - sin ( 6 x )} = - cot (x)