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شائع علم المثلثات >

prove (sec(α)+csc(α))(cos(α)-sin(α))=cot(α)-tan(α)

الحلّ

prove

(sec (α) + csc (α)) (cos (α) - sin (α)) = cot (α) - tan (α)

الحلّ

صحيح

خطوات الحلّ

(sec (α) + csc (α)) (cos (α) - sin (α)) = cot (α) - tan (α)

قم بالعمل على الطرف الأيسر

(sec (α) + csc (α)) (cos (α) - sin (α))

sin, cos :

عبّر بواسطة

(cos (α) - sin (α)) (csc (α) + sec (α))

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

:Use the basic trigonometric identity

= (cos (α) - sin (α)) (\frac{1}{sin ( α )} + sec (α))

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

:Use the basic trigonometric identity

= (cos (α) - sin (α)) (\frac{1}{sin ( α )} + \frac{1}{cos ( α )})

(cos (α) - sin (α)) (\frac{1}{sin ( α )} + \frac{1}{cos ( α )})

بسّط:

\frac{( cos ( α ) + sin ( α ) ) ( cos ( α ) - sin ( α ) )}{sin ( α ) cos ( α )}

(cos (α) - sin (α)) (\frac{1}{sin ( α )} + \frac{1}{cos ( α )})

\frac{1}{sin ( α )} + \frac{1}{cos ( α )}

وحّد:

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

\frac{1}{sin ( α )} + \frac{1}{cos ( α )}

sin (α), cos (α)

المضاعف المشترك الأصغر لـ:

sin (α) cos (α)

sin (α), cos (α)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (α)

cos (α)

= sin (α) cos (α)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (α) cos (α)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{1}{sin ( α )} :

multiply the denominator and numerator by

cos (α)

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

For

\frac{1}{cos ( α )} :

multiply the denominator and numerator by

sin (α)

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

= \frac{cos ( α )}{sin ( α ) cos ( α )} + \frac{sin ( α )}{sin ( α ) cos ( α )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{cos ( α ) + sin ( α )}{sin ( α ) cos ( α )}

= \frac{cos ( α ) + sin ( α )}{sin ( α ) cos ( α )} (cos (α) - sin (α))

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

:اضرب كسور

= \frac{( cos ( α ) + sin ( α ) ) ( cos ( α ) - sin ( α ) )}{sin ( α ) cos ( α )}

= \frac{( cos ( α ) + sin ( α ) ) ( cos ( α ) - sin ( α ) )}{sin ( α ) cos ( α )}

= \frac{( cos ( α ) + sin ( α ) ) ( cos ( α ) - sin ( α ) )}{cos ( α ) sin ( α )}

(cos (α) + sin (α)) (cos (α) - sin (α))

وسٌع:

cos^{2} (α) - sin^{2} (α)

(cos (α) + sin (α)) (cos (α) - sin (α))

(a + b) (a - b) = a^{2} - b^{2}

فعّل قانون فرق المربّعات

a = cos (α), b = sin (α)

= cos^{2} (α) - sin^{2} (α)

= \frac{cos ^{2} ( α ) - sin ^{2} ( α )}{cos ( α ) sin ( α )}

قم بالعمل على الطرف الأيمن

cot (α) - tan (α)

sin, cos :

عبّر بواسطة

cot (α) - tan (α)

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

:Use the basic trigonometric identity

= \frac{cos ( α )}{sin ( α )} - tan (α)

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

:Use the basic trigonometric identity

= \frac{cos ( α )}{sin ( α )} - \frac{sin ( α )}{cos ( α )}

\frac{cos ( α )}{sin ( α )} - \frac{sin ( α )}{cos ( α )}

بسّط:

\frac{cos ^{2} ( α ) - sin ^{2} ( α )}{sin ( α ) cos ( α )}

\frac{cos ( α )}{sin ( α )} - \frac{sin ( α )}{cos ( α )}

sin (α), cos (α)

المضاعف المشترك الأصغر لـ:

sin (α) cos (α)

sin (α), cos (α)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (α)

cos (α)

= sin (α) cos (α)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (α) cos (α)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{cos ( α )}{sin ( α )} :

multiply the denominator and numerator by

cos (α)

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

For

\frac{sin ( α )}{cos ( α )} :

multiply the denominator and numerator by

sin (α)

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

= \frac{cos ^{2} ( α )}{sin ( α ) cos ( α )} - \frac{sin ^{2} ( α )}{sin ( α ) cos ( α )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{cos ^{2} ( α ) - sin ^{2} ( α )}{sin ( α ) cos ( α )}

= \frac{cos ^{2} ( α ) - sin ^{2} ( α )}{sin ( α ) cos ( α )}

= \frac{cos ^{2} ( α ) - sin ^{2} ( α )}{cos ( α ) sin ( α )}

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

أمثلة شائعة

prove 1-(-cos(x))^2=1+cos^2(x)

p ro v e 1 - (- cos (x))^{2} = 1 + cos^{2} (x)

prove csc^2(x/2)=(2sec(x))/(sec(x)-1)

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

prove (sec(x))/(-csc(x))+(-sin(x))/(cos(x))=-2tan(x)

p ro v e \frac{sec ( x )}{- csc ( x )} + \frac{- sin ( x )}{cos ( x )} = - 2 tan (x)

prove 1=(sec(u)+tan(u))/(sec(u)+tan(u))

p ro v e 1 = \frac{sec ( u ) + tan ( u )}{sec ( u ) + tan ( u )}

prove (sin^2(x))/1*1/(sin^2(x))=1

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