English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

شائع علم المثلثات >

prove 7(cot^2(x))/(csc(x))sec^2(x)=7tan(x)cos(x)csc^2(x)

الحلّ

prove

7 \frac{cot ^{2} ( x )}{csc ( x )} sec^{2} (x) = 7 tan (x) cos (x) csc^{2} (x)

الحلّ

صحيح

خطوات الحلّ

7 \cdot \frac{cot ^{2} ( x )}{csc ( x )} sec^{2} (x) = 7 tan (x) cos (x) csc^{2} (x)

قم بالعمل على الطرف الأيسر

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

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

= 7 \cdot \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{csc ( x )} sec^{2} (x)

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

:Use the basic trigonometric identity

= 7 \cdot \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{\frac{1}{s i n ( x )}} sec^{2} (x)

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

:Use the basic trigonometric identity

= 7 \cdot \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{\frac{1}{s i n ( x )}} (\frac{1}{cos ( x )})^{2}

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

بسّط:

\frac{7}{sin ( x )}

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

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

:اضرب كسور

= \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2} \cdot 7 ( \frac{1}{c o s ( x )} ) ^{2}}{\frac{1}{s i n ( x )}}

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

: استخدم ميزات الكسور التالية

= \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2} \cdot 7 ( \frac{1}{c o s ( x )} ) ^{2} sin ( x )}{1}

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

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

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

:فعّل قانون القوى

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

= \frac{7 ( \frac{1}{c o s ( x )} ) ^{2} \frac{c o s ^{2} ( x )}{s i n ^{2} ( x )} sin ( x )}{1}

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

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

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

:فعّل قانون القوى

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

= \frac{7 \cdot \frac{c o s ^{2} ( x )}{s i n ^{2} ( x )} \cdot \frac{1}{c o s ^{2} ( x )} sin ( x )}{1}

\frac{cos ^{2} ( x )}{sin ^{2} ( x )} \cdot 7 \cdot \frac{1}{cos ^{2} ( x )} sin (x)

اضرب بـ:

\frac{7}{sin ( x )}

\frac{cos ^{2} ( x )}{sin ^{2} ( x )} \cdot 7 \cdot \frac{1}{cos ^{2} ( x )} sin (x)

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

:اضرب كسور

= \frac{cos ^{2} ( x ) \cdot 1 \cdot 7 sin ( x )}{sin ^{2} ( x ) cos ^{2} ( x )}

cos^{2} (x) :

إلغ العوامل المشتركة

= \frac{1 \cdot 7 sin ( x )}{sin ^{2} ( x )}

1 \cdot 7 = 7 :

اضرب الأعداد

= \frac{7 sin ( x )}{sin ^{2} ( x )}

sin (x) :

إلغ العوامل المشتركة

= \frac{7}{sin ( x )}

= \frac{\frac{7}{s i n ( x )}}{1}

\frac{a}{1} = a

: استخدم ميزات الكسور التالية

= \frac{7}{sin ( x )}

= \frac{7}{sin ( x )}

= \frac{7}{sin ( x )}

قم بالعمل على الطرف الأيمن

7 tan (x) cos (x) csc^{2} (x)

sin, cos :

عبّر بواسطة

7 cos (x) csc^{2} (x) tan (x)

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

:Use the basic trigonometric identity

= 7 cos (x) (\frac{1}{sin ( x )})^{2} tan (x)

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

:Use the basic trigonometric identity

= 7 cos (x) (\frac{1}{sin ( x )})^{2} \frac{sin ( x )}{cos ( x )}

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

بسّط:

\frac{7}{sin ( x )}

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

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

:اضرب كسور

= \frac{sin ( x ) \cdot 7 cos ( x ) ( \frac{1}{s i n ( x )} ) ^{2}}{cos ( x )}

cos (x) :

إلغ العوامل المشتركة

= sin (x) \cdot 7 (\frac{1}{sin ( x )})^{2}

(\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 )}

= 7 \cdot \frac{1}{sin ^{2} ( x )} sin (x)

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

:اضرب كسور

= \frac{1 \cdot sin ( x ) \cdot 7}{sin ^{2} ( x )}

1 \cdot 7 = 7 :

اضرب الأعداد

= \frac{7 sin ( x )}{sin ^{2} ( x )}

sin (x) :

إلغ العوامل المشتركة

= \frac{7}{sin ( x )}

= \frac{7}{sin ( x )}

= \frac{7}{sin ( x )}

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

أمثلة شائعة

prove cos^2(x)=1+sin^2(x)

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

prove cos^2(x)(2+tan^2(x))=2-sin^2(x)

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

prove sin(x)(1+cos(2x))=sin(2x)cos(x)

p ro v e sin (x) (1 + cos (2 x)) = sin (2 x) cos (x)

prove (sec(x)-tan(x))(1+sin(x))=cos(x)

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

prove tan(-x)=cos(x)

p ro v e tan (- x) = cos (x)