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العربية

prove ((csc^2(x)))/(cot(x))=csc(x)sec(x)

الحلّ

prove

\frac{( csc ^{2} ( x ) )}{cot ( x )} = csc (x) sec (x)

صحيح

خطوات الحلّ

\frac{csc ^{2} ( x )}{cot ( x )} = csc (x) sec (x)

قم بالعمل على الطرف الأيسر

\frac{csc ^{2} ( x )}{cot ( x )}

sin, cos :

عبّر بواسطة

\frac{csc ^{2} ( x )}{cot ( x )}

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

:Use the basic trigonometric identity

= \frac{( \frac{1}{s i n ( x )} ) ^{2}}{cot ( x )}

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

:Use the basic trigonometric identity

= \frac{( \frac{1}{s i n ( x )} ) ^{2}}{\frac{c o s ( x )}{s i n ( x )}}

\frac{( \frac{1}{s i n ( x )} ) ^{2}}{\frac{c o s ( x )}{s i n ( x )}}

بسّط:

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

\frac{( \frac{1}{s i n ( x )} ) ^{2}}{\frac{c o s ( x )}{s i n ( x )}}

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

: استخدم ميزات الكسور التالية

= \frac{( \frac{1}{s i n ( x )} ) ^{2} sin ( x )}{cos ( x )}

(\frac{1}{sin ( x )})^{2} = \frac{1}{sin ^{2} ( x )}

(\frac{1}{sin ( x )})^{2}

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

:فعّل قانون القوى

= \frac{1 ^{2}}{sin ^{2} ( x )}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{sin ^{2} ( x )}

= \frac{\frac{1}{s i n ^{2} ( x )} sin ( x )}{cos ( x )}

\frac{1}{sin ^{2} ( x )} sin (x)

اضرب بـ:

\frac{1}{sin ( x )}

\frac{1}{sin ^{2} ( x )} sin (x)

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

:اضرب كسور

= \frac{1 \cdot sin ( x )}{sin ^{2} ( x )}

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

اضرب

= \frac{sin ( x )}{sin ^{2} ( x )}

sin (x) :

إلغ العوامل المشتركة

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

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

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

: استخدم ميزات الكسور التالية

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

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

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

:Use the basic trigonometric identity

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

بسّط

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

1 \cdot 1 = 1 :

اضرب الأعداد

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

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

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

: استخدم ميزات الكسور التالية

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

\frac{a}{1} = a

فعّل القانون

= sec (x) csc (x)

sec (x) csc (x)

sec (x) csc (x)

= csc (x) sec (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e csc^{2} (x) (tan^{2} (x)) - 1 = tan^{2} (x)

p ro v e sin (x) cos (x) + sin (3 x) sec (x) = tan (x)

p ro v e (1 + sec (x)) (1 - cos (x)) = sin (x) tan (x)

p ro v e sec (x) \cdot sin (x) = tan (x)

p ro v e tan (x) csc (x) sec (x) = tan^{2} (x) + 1