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العربية

prove (sec(x)csc(x))/2 =csc(2x)

الحلّ

prove

\frac{sec ( x ) csc ( x )}{2} = csc (2 x)

صحيح

خطوات الحلّ

\frac{sec ( x ) csc ( x )}{2} = csc (2 x)

قم بالعمل على الطرف الأيمن

csc (2 x)

sin, cos :

عبّر بواسطة

csc (2 x)

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

:Use the basic trigonometric identity

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

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

Rewrite using trig identities

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

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

:فعّل متطابقة الزاوية المضاعفة

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

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

Rewrite using trig identities

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

:Use the basic trigonometric identity

\frac{1}{2 \cdot \frac{1}{c s c ( x )} cos ( x )}

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

:Use the basic trigonometric identity

\frac{1}{2 \cdot \frac{1}{c s c ( x )} \cdot \frac{1}{s e c ( x )}}

بسّط

\frac{1}{2 \cdot \frac{1}{c s c ( x )} \cdot \frac{1}{s e c ( x )}}

2 \cdot \frac{1}{csc ( x )} \cdot \frac{1}{sec ( x )}

اضرب بـ:

\frac{2}{csc ( x ) sec ( x )}

2 \cdot \frac{1}{csc ( x )} \cdot \frac{1}{sec ( x )}

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

:اضرب كسور

= \frac{1 \cdot 1 \cdot 2}{csc ( x ) sec ( x )}

1 \cdot 1 \cdot 2 = 2 :

اضرب الأعداد

= \frac{2}{csc ( x ) sec ( x )}

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

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

: استخدم ميزات الكسور التالية

= \frac{csc ( x ) sec ( x )}{2}

\frac{csc ( x ) sec ( x )}{2}

\frac{csc ( x ) sec ( x )}{2}

= \frac{sec ( x ) csc ( x )}{2}

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e - sec^{2} (x) = - tan^{2} (x) - 1

p ro v e tan (4 x) \cdot cos (4 x) = sin (4 x)

p ro v e tan (θ) = csc (θ) sec (θ) - cot (θ)

p ro v e sec (x) - sec (x) sin (x^{2}) = cos (x^{2})

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