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العربية

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

الحلّ

prove

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

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

2 csc (2 x)

sin, cos :

عبّر بواسطة

2 csc (2 x)

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

:اضرب كسور

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

1 \cdot 2 = 2 :

اضرب الأعداد

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

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

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

Rewrite using trig identities

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

sin (2 x) = 2 sin (x) cos (x)

:فعّل متطابقة الزاوية المضاعفة

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

\frac{2}{2} = 1 :

اقسم الأعداد

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

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

قم بالعمل على الطرف الأيمن

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

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

:اضرب كسور

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

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

اضرب بـ:

\frac{1}{cos ( x )}

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

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

:اضرب كسور

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

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

اضرب

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

cos (x) :

إلغ العوامل المشتركة

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

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

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

: استخدم ميزات الكسور التالية

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

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

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

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e sin^{2} (2 θ) = \frac{1 - cos ( 4 θ )}{2}

p ro v e \frac{1}{tan ( x )} = arctan (x)

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

p ro v e \frac{( sec ( θ ) - tan ( θ ) ) ^{2} + 1}{csc ( θ ) ( sec ( θ ) - tan ( θ ) )} = 2 tan (θ)

p ro v e 2 csc (2 x) = sec (x) csc (x)