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prove csc(θ)=cot(θ)sec(θ)

الحلّ

prove

csc (θ) = cot (θ) sec (θ)

صحيح

خطوات الحلّ

csc (θ) = cot (θ) sec (θ)

قم بالعمل على الطرف الأيمن

cot (θ) sec (θ)

sin, cos :

عبّر بواسطة

cot (θ) sec (θ)

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

:Use the basic trigonometric identity

= \frac{cos ( θ )}{sin ( θ )} sec (θ)

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

:Use the basic trigonometric identity

= \frac{cos ( θ )}{sin ( θ )} \cdot \frac{1}{cos ( θ )}

\frac{cos ( θ )}{sin ( θ )} \cdot \frac{1}{cos ( θ )}

بسّط:

\frac{1}{sin ( θ )}

\frac{cos ( θ )}{sin ( θ )} \cdot \frac{1}{cos ( θ )}

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

:اضرب كسور

= \frac{cos ( θ ) \cdot 1}{sin ( θ ) cos ( θ )}

cos (θ) :

إلغ العوامل المشتركة

= \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )}

Rewrite using trig identities

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

:Use the basic trigonometric identity

\frac{1}{\frac{1}{c s c ( θ )}}

بسّط

\frac{1}{\frac{1}{c s c ( θ )}}

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

: استخدم ميزات الكسور التالية

= \frac{csc ( θ )}{1}

\frac{a}{1} = a

فعّل القانون

= csc (θ)

csc (θ)

csc (θ)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

\frac{1 - tan ( x )}{sin ( x )} = csc (x) - sec (x)

sec (θ) sin (θ) cot (θ) = 1

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

\frac{tan ^{2} ( θ )}{sin ^{2} ( θ )} - 1 = tan^{2} (θ)

csc (θ) cos^{2} (θ) + sin (θ) = csc (θ)