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prove (sec(a))/(tan(a)+cot(a))=sin(a)

الحلّ

prove

\frac{sec ( a )}{tan ( a ) + cot ( a )} = sin (a)

الحلّ

صحيح

خطوات الحلّ

\frac{sec ( a )}{tan ( a ) + cot ( a )} = sin (a)

قم بالعمل على الطرف الأيسر

\frac{sec ( a )}{tan ( a ) + cot ( a )}

sin, cos :

عبّر بواسطة

\frac{sec ( a )}{cot ( a ) + tan ( a )}

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

:Use the basic trigonometric identity

= \frac{\frac{1}{c o s ( a )}}{cot ( a ) + tan ( a )}

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

: استخدم ميزات الكسور التالية

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

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

وحّد:

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

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

sin (a), cos (a)

المضاعف المشترك الأصغر لـ:

sin (a) cos (a)

sin (a), cos (a)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (a)

cos (a)

= sin (a) cos (a)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (a) cos (a)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{cos ( a )}{sin ( a )} :

multiply the denominator and numerator by

cos (a)

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

For

\frac{sin ( a )}{cos ( a )} :

multiply the denominator and numerator by

sin (a)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

cos (a) :

إلغ العوامل المشتركة

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

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

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

: استخدم ميزات الكسور التالية

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

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

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