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prove tan(x)*csc^2(x)-tan(x)=cot(x)

الحلّ

prove

tan (x) \cdot csc^{2} (x) - tan (x) = cot (x)

الحلّ

صحيح

خطوات الحلّ

tan (x) csc^{2} (x) - tan (x) = cot (x)

قم بالعمل على الطرف الأيسر

tan (x) csc^{2} (x) - tan (x)

sin, cos :

عبّر بواسطة

- tan (x) + csc^{2} (x) tan (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

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

اضرب بـ:

\frac{1}{sin ( x )}

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

cos (x), sin (x) cos (x)

المضاعف المشترك الأصغر لـ:

cos (x) sin (x)

cos (x), sin (x) cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (x)

sin (x) cos (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{sin ( x )}{cos ( x )} :

multiply the denominator and numerator by

sin (x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

Rewrite using trig identities

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

1 = cos^{2} (x) + sin^{2} (x)

:فعّل نطريّة فيتاغوروس

1 - sin^{2} (x) = cos^{2} (x)

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

cos (x) :

إلغ العوامل المشتركة

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

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

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

:Use the basic trigonometric identity

= cot (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

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