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עברית

العربية

prove tan(x)sec^4(x)=(sin(x))/(cos^5(x))

الحلّ

prove

tan (x) sec^{4} (x) = \frac{sin ( x )}{cos ^{5} ( x )}

صحيح

خطوات الحلّ

tan (x) sec^{4} (x) = \frac{sin ( x )}{cos ^{5} ( x )}

قم بالعمل على الطرف الأيسر

tan (x) sec^{4} (x)

sin, cos :

عبّر بواسطة

sec^{4} (x) tan (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

\frac{sin ( x )}{cos ^{5} ( x )}

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

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

:اضرب كسور

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

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

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

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

:فعّل قانون القوى

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

1^{a} = 1

فعّل القانون

1^{4} = 1

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

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

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

اضرب بـ:

\frac{sin ( x )}{cos ^{4} ( x )}

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

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

:اضرب كسور

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

1 \cdot sin (x) = sin (x) :

اضرب

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

= \frac{\frac{s i n ( x )}{c o s ^{4} ( x )}}{cos ( x )}

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

: استخدم ميزات الكسور التالية

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

cos^{4} (x) cos (x) = cos^{5} (x)

cos^{4} (x) cos (x)

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

cos^{4} (x) cos (x) = cos^{4 + 1} (x)

= cos^{4 + 1} (x)

4 + 1 = 5 :

اجمع الأعداد

= cos^{5} (x)

= \frac{sin ( x )}{cos ^{5} ( x )}

= \frac{sin ( x )}{cos ^{5} ( x )}

= \frac{sin ( x )}{cos ^{5} ( x )}

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e csc (x) + cot (x) sec (x) - 1 = tan (x)

p ro v e \frac{3 csc ( x ) - 3 sin ( x )}{tan ( x ) - cot ( x )} = 3 cos^{3} (x)

p ro v e 9 cos (x^{\circ}) + 6 sin (x^{\circ}) = 10

p ro v e \frac{tan ^{2} ( A )}{sec ^{2} ( A )} = sin^{2} (A)

p ro v e 5 cos^{2} (x) - 2 cos (x) - 3 - sin^{2} (x) = 0