English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove (csc(a))/(1+cot^2(a))=sin(a)

الحلّ

prove

\frac{csc ( a )}{1 + cot ^{2} ( a )} = sin (a)

صحيح

خطوات الحلّ

\frac{csc ( a )}{1 + cot ^{2} ( a )} = sin (a)

قم بالعمل على الطرف الأيسر

\frac{csc ( a )}{1 + cot ^{2} ( a )}

sin, cos :

عبّر بواسطة

\frac{csc ( a )}{1 + cot ^{2} ( a )}

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

:Use the basic trigonometric identity

= \frac{\frac{1}{s i n ( a )}}{1 + cot ^{2} ( a )}

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

:Use the basic trigonometric identity

= \frac{\frac{1}{s i n ( a )}}{1 + ( \frac{c o s ( a )}{s i n ( a )} ) ^{2}}

\frac{\frac{1}{s i n ( a )}}{1 + ( \frac{c o s ( a )}{s i n ( a )} ) ^{2}}

بسّط:

\frac{sin ( a )}{sin ^{2} ( a ) + cos ^{2} ( a )}

\frac{\frac{1}{s i n ( a )}}{1 + ( \frac{c o s ( a )}{s i n ( a )} ) ^{2}}

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

: استخدم ميزات الكسور التالية

= \frac{1}{sin ( a ) ( 1 + ( \frac{c o s ( a )}{s i n ( a )} ) ^{2} )}

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

:فعّل قانون القوى

= \frac{1}{sin ( a ) ( \frac{c o s ^{2} ( a )}{s i n ^{2} ( a )} + 1 )}

1 + \frac{cos ^{2} ( a )}{sin ^{2} ( a )}

وحّد:

\frac{sin ^{2} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

1 + \frac{cos ^{2} ( a )}{sin ^{2} ( a )}

1 = \frac{1 sin ^{2} ( a )}{sin ^{2} ( a )}

:حوّل الأعداد لكسور

= \frac{1 \cdot sin ^{2} ( a )}{sin ^{2} ( a )} + \frac{cos ^{2} ( a )}{sin ^{2} ( a )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{1 \cdot sin ^{2} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

1 \cdot sin^{2} (a) = sin^{2} (a) :

اضرب

= \frac{sin ^{2} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

= \frac{1}{\frac{s i n ^{2} ( a ) + c o s ^{2} ( a )}{s i n ^{2} ( a )} sin ( a )}

sin (a) \frac{sin ^{2} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

اضرب بـ:

\frac{sin ^{2} ( a ) + cos ^{2} ( a )}{sin ( a )}

sin (a) \frac{sin ^{2} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

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

:اضرب كسور

= \frac{( sin ^{2} ( a ) + cos ^{2} ( a ) ) sin ( a )}{sin ^{2} ( a )}

sin (a) :

إلغ العوامل المشتركة

= \frac{sin ^{2} ( a ) + cos ^{2} ( a )}{sin ( a )}

= \frac{1}{\frac{s i n ^{2} ( a ) + c o s ^{2} ( a )}{s i n ( a )}}

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

: استخدم ميزات الكسور التالية

= \frac{sin ( a )}{sin ^{2} ( a ) + cos ^{2} ( a )}

= \frac{sin ( a )}{sin ^{2} ( a ) + cos ^{2} ( a )}

= \frac{sin ( a )}{cos ^{2} ( a ) + sin ^{2} ( a )}

Rewrite using trig identities

\frac{sin ( a )}{cos ^{2} ( a ) + sin ^{2} ( a )}

cos^{2} (x) + sin^{2} (x) = 1

:فعّل نطريّة فيتاغوروس

= \frac{sin ( a )}{1}

\frac{a}{1} = a

فعّل القانون

= sin (a)

= sin (a)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

\frac{1}{tan ( x )} = arctan (x)

1 + cos^{2} (x) = 2 cos^{2} (x) + sin^{2} (x)

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

csc (x) = 2

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