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شائع علم المثلثات >

prove 9/(tan(x))+9/(cot(x))=9tan(x)+9cot(x)

الحلّ

prove

\frac{9}{tan ( x )} + \frac{9}{cot ( x )} = 9 tan (x) + 9 cot (x)

الحلّ

صحيح

خطوات الحلّ

\frac{9}{tan ( x )} + \frac{9}{cot ( x )} = 9 tan (x) + 9 cot (x)

قم بالعمل على الطرف الأيسر

\frac{9}{tan ( x )} + \frac{9}{cot ( x )}

sin, cos :

عبّر بواسطة

\frac{9}{cot ( x )} + \frac{9}{tan ( x )}

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

:Use the basic trigonometric identity

= \frac{9}{\frac{c o s ( x )}{s i n ( x )}} + \frac{9}{tan ( x )}

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

:Use the basic trigonometric identity

= \frac{9}{\frac{c o s ( x )}{s i n ( x )}} + \frac{9}{\frac{s i n ( x )}{c o s ( x )}}

\frac{9}{\frac{c o s ( x )}{s i n ( x )}} + \frac{9}{\frac{s i n ( x )}{c o s ( x )}}

بسّط:

\frac{9 sin ^{2} ( x ) + 9 cos ^{2} ( x )}{cos ( x ) sin ( x )}

\frac{9}{\frac{c o s ( x )}{s i n ( x )}} + \frac{9}{\frac{s i n ( x )}{c o s ( x )}}

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

: استخدم ميزات الكسور التالية

= \frac{9 sin ( x )}{cos ( x )} + \frac{9}{\frac{s i n ( x )}{c o s ( x )}}

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

: استخدم ميزات الكسور التالية

= \frac{9 sin ( x )}{cos ( x )} + \frac{9 cos ( x )}{sin ( x )}

cos (x), sin (x)

المضاعف المشترك الأصغر لـ:

cos (x) sin (x)

cos (x), sin (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (x)

sin (x)

= cos (x) sin (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

cos (x) sin (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{9 sin ( x )}{cos ( x )} :

multiply the denominator and numerator by

sin (x)

\frac{9 sin ( x )}{cos ( x )} = \frac{9 sin ( x ) sin ( x )}{cos ( x ) sin ( x )} = \frac{9 sin ^{2} ( x )}{cos ( x ) sin ( x )}

For

\frac{9 cos ( x )}{sin ( x )} :

multiply the denominator and numerator by

cos (x)

\frac{9 cos ( x )}{sin ( x )} = \frac{9 cos ( x ) cos ( x )}{sin ( x ) cos ( x )} = \frac{9 cos ^{2} ( x )}{cos ( x ) sin ( x )}

= \frac{9 sin ^{2} ( x )}{cos ( x ) sin ( x )} + \frac{9 cos ^{2} ( x )}{cos ( x ) sin ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{9 sin ^{2} ( x ) + 9 cos ^{2} ( x )}{cos ( x ) sin ( x )}

= \frac{9 sin ^{2} ( x ) + 9 cos ^{2} ( x )}{cos ( x ) sin ( x )}

= \frac{9 cos ^{2} ( x ) + 9 sin ^{2} ( x )}{cos ( x ) sin ( x )}

قم بالعمل على الطرف الأيمن

9 tan (x) + 9 cot (x)

sin, cos :

عبّر بواسطة

9 cot (x) + 9 tan (x)

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

:Use the basic trigonometric identity

= 9 \cdot \frac{cos ( x )}{sin ( x )} + 9 tan (x)

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

:Use the basic trigonometric identity

= 9 \cdot \frac{cos ( x )}{sin ( x )} + 9 \cdot \frac{sin ( x )}{cos ( x )}

9 \cdot \frac{cos ( x )}{sin ( x )} + 9 \cdot \frac{sin ( x )}{cos ( x )}

بسّط:

\frac{9 cos ^{2} ( x ) + 9 sin ^{2} ( x )}{sin ( x ) cos ( x )}

9 \cdot \frac{cos ( x )}{sin ( x )} + 9 \cdot \frac{sin ( x )}{cos ( x )}

9 \cdot \frac{cos ( x )}{sin ( x )}

اضرب بـ:

\frac{9 cos ( x )}{sin ( x )}

9 \cdot \frac{cos ( x )}{sin ( x )}

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

:اضرب كسور

= \frac{cos ( x ) \cdot 9}{sin ( x )}

= \frac{9 cos ( x )}{sin ( x )} + 9 \cdot \frac{sin ( x )}{cos ( x )}

9 \cdot \frac{sin ( x )}{cos ( x )}

اضرب بـ:

\frac{9 sin ( x )}{cos ( x )}

9 \cdot \frac{sin ( x )}{cos ( x )}

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

:اضرب كسور

= \frac{sin ( x ) \cdot 9}{cos ( x )}

= \frac{9 cos ( x )}{sin ( x )} + \frac{9 sin ( x )}{cos ( x )}

sin (x), cos (x)

المضاعف المشترك الأصغر لـ:

sin (x) cos (x)

sin (x), cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

cos (x)

= sin (x) cos (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (x) cos (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{cos ( x ) \cdot 9}{sin ( x )} :

multiply the denominator and numerator by

cos (x)

\frac{cos ( x ) \cdot 9}{sin ( x )} = \frac{cos ( x ) \cdot 9 cos ( x )}{sin ( x ) cos ( x )} = \frac{9 cos ^{2} ( x )}{sin ( x ) cos ( x )}

For

\frac{sin ( x ) \cdot 9}{cos ( x )} :

multiply the denominator and numerator by

sin (x)

\frac{sin ( x ) \cdot 9}{cos ( x )} = \frac{sin ( x ) \cdot 9 sin ( x )}{cos ( x ) sin ( x )} = \frac{9 sin ^{2} ( x )}{sin ( x ) cos ( x )}

= \frac{9 cos ^{2} ( x )}{sin ( x ) cos ( x )} + \frac{9 sin ^{2} ( x )}{sin ( x ) cos ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{9 cos ^{2} ( x ) + 9 sin ^{2} ( x )}{sin ( x ) cos ( x )}

= \frac{9 cos ^{2} ( x ) + 9 sin ^{2} ( x )}{sin ( x ) cos ( x )}

= \frac{9 cos ^{2} ( x ) + 9 sin ^{2} ( x )}{cos ( x ) sin ( x )}

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

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