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العربية

prove 1/(sec^3(x)cos^4(x))=sec(x)

الحلّ

prove

\frac{1}{sec ^{3} ( x ) cos ^{4} ( x )} = sec (x)

صحيح

خطوات الحلّ

\frac{1}{sec ^{3} ( x ) cos ^{4} ( x )} = sec (x)

قم بالعمل على الطرف الأيسر

\frac{1}{sec ^{3} ( x ) cos ^{4} ( x )}

sin, cos :

عبّر بواسطة

\frac{1}{cos ^{4} ( x ) sec ^{3} ( x )}

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

:Use the basic trigonometric identity

= \frac{1}{cos ^{4} ( x ) ( \frac{1}{c o s ( x )} ) ^{3}}

\frac{1}{cos ^{4} ( x ) ( \frac{1}{c o s ( x )} ) ^{3}}

بسّط:

\frac{1}{cos ( x )}

\frac{1}{cos ^{4} ( x ) ( \frac{1}{c o s ( x )} ) ^{3}}

(\frac{1}{cos ( x )})^{3} = \frac{1}{cos ^{3} ( x )}

(\frac{1}{cos ( x )})^{3}

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

:فعّل قانون القوى

= \frac{1 ^{3}}{cos ^{3} ( x )}

1^{a} = 1

فعّل القانون

1^{3} = 1

= \frac{1}{cos ^{3} ( x )}

= \frac{1}{\frac{1}{c o s ^{3} ( x )} cos ^{4} ( x )}

cos^{4} (x) \frac{1}{cos ^{3} ( x )}

اضرب بـ:

cos (x)

cos^{4} (x) \frac{1}{cos ^{3} ( x )}

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

:اضرب كسور

= \frac{1 \cdot cos ^{4} ( x )}{cos ^{3} ( x )}

1 \cdot cos^{4} (x) = cos^{4} (x) :

اضرب

= \frac{cos ^{4} ( x )}{cos ^{3} ( x )}

\frac{x ^{a}}{x ^{b}} = x^{a - b}

:فعّل قانون القوى

\frac{cos ^{4} ( x )}{cos ^{3} ( x )} = cos^{4 - 3} (x)

= cos^{4 - 3} (x)

4 - 3 = 1 :

اطرح الأعداد

= cos (x)

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

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

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

Rewrite using trig identities

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

:Use the basic trigonometric identity

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

بسّط

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

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

: استخدم ميزات الكسور التالية

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

\frac{a}{1} = a

فعّل القانون

= sec (x)

sec (x)

sec (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e csc^{2} (θ) + 1 = cot^{2} (θ)

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

p ro v e cos^{2} (7 θ) - sin^{2} (7 θ) = cos (14 θ)

p ro v e sin^{4} (x) - (\frac{3}{4 \cdot sin ^{2} ( x )}) + 1 = 1

p ro v e arccot (x) = tan (x)