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العربية

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

الحلّ

prove

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

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

cot^{2} (x) sec^{2} (x)

sin, cos :

عبّر بواسطة

cot^{2} (x) sec^{2} (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

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

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

:اضرب كسور

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

cos^{2} (x) :

إلغ العوامل المشتركة

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

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

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

Rewrite using trig identities

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

:Use the basic trigonometric identity

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

بسّط

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

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

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

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

:فعّل قانون القوى

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

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

: استخدم ميزات الكسور التالية

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

\frac{a}{1} = a

فعّل القانون

= csc^{2} (x)

csc^{2} (x)

csc^{2} (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e cot (\frac{1}{2} x) = \frac{sin ( x )}{1 - cos ( x )}

p ro v e cos (π - t) = - cos (t)

p ro v e sin (\frac{θ}{2}) = 12 - 20

p ro v e cos^{3} (x) + cos (x) sin^{2} (x) = cos (x)

p ro v e sec (x) + tan (x) = \frac{1}{( sec ( x ) - tan ( x ) )}