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prove sec(t)(csc(t)(tan(t)+cot(t)))=sec^2(t)+csc^2(t)

الحلّ

prove

sec (t) (csc (t) (tan (t) + cot (t))) = sec^{2} (t) + csc^{2} (t)

الحلّ

صحيح

خطوات الحلّ

sec (t) csc (t) (tan (t) + cot (t)) = sec^{2} (t) + csc^{2} (t)

قم بالعمل على الطرف الأيسر

sec (t) csc (t) (tan (t) + cot (t))

sin, cos :

عبّر بواسطة

(cot (t) + tan (t)) csc (t) sec (t)

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

:Use the basic trigonometric identity

= (\frac{cos ( t )}{sin ( t )} + tan (t)) csc (t) sec (t)

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

:Use the basic trigonometric identity

= (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) csc (t) sec (t)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

= (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) \frac{1}{sin ( t )} \cdot \frac{1}{cos ( t )}

(\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) \frac{1}{sin ( t )} \cdot \frac{1}{cos ( t )}

بسّط:

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

(\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) \frac{1}{sin ( t )} \cdot \frac{1}{cos ( t )}

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 ( \frac{c o s ( t )}{s i n ( t )} + \frac{s i n ( t )}{c o s ( t )} )}{sin ( t ) cos ( t )}

1 \cdot 1 \cdot (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) = \frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}

1 \cdot 1 \cdot (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )})

1 \cdot 1 \cdot (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) = (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) :

اضرب

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

(a) = a

:احذف الأقواس

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

= \frac{\frac{c o s ( t )}{s i n ( t )} + \frac{s i n ( t )}{c o s ( t )}}{sin ( t ) cos ( t )}

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

وحّد:

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

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

sin (t), cos (t)

المضاعف المشترك الأصغر لـ:

sin (t) cos (t)

sin (t), cos (t)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (t)

cos (t)

= sin (t) cos (t)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (t) cos (t)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

cos (t)

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

For

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

multiply the denominator and numerator by

sin (t)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

: استخدم ميزات الكسور التالية

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

sin (t) cos (t) sin (t) cos (t) = sin^{2} (t) cos^{2} (t)

sin (t) cos (t) sin (t) cos (t)

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

cos (t) cos (t) = cos^{1 + 1} (t)

= sin (t) sin (t) cos^{1 + 1} (t)

1 + 1 = 2 :

اجمع الأعداد

= sin (t) sin (t) cos^{2} (t)

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

sin (t) sin (t) = sin^{1 + 1} (t)

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

1 + 1 = 2 :

اجمع الأعداد

= sin^{2} (t) cos^{2} (t)

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

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

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

Rewrite using trig identities

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

:Use the basic trigonometric identity

\frac{cos ^{2} ( t ) + ( \frac{1}{c s c ( t )} ) ^{2}}{cos ^{2} ( t ) ( \frac{1}{c s c ( t )} ) ^{2}}

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

:Use the basic trigonometric identity

\frac{( \frac{1}{s e c ( t )} ) ^{2} + ( \frac{1}{c s c ( t )} ) ^{2}}{( \frac{1}{s e c ( t )} ) ^{2} ( \frac{1}{c s c ( t )} ) ^{2}}

بسّط

\frac{( \frac{1}{s e c ( t )} ) ^{2} + ( \frac{1}{c s c ( t )} ) ^{2}}{( \frac{1}{s e c ( t )} ) ^{2} ( \frac{1}{c s c ( t )} ) ^{2}}

(\frac{1}{sec ( t )})^{2} = \frac{1}{sec ^{2} ( t )}

(\frac{1}{sec ( t )})^{2}

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

:فعّل قانون القوى

= \frac{1 ^{2}}{sec ^{2} ( t )}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{sec ^{2} ( t )}

= \frac{( \frac{1}{s e c ( t )} ) ^{2} + ( \frac{1}{c s c ( t )} ) ^{2}}{( \frac{1}{c s c ( t )} ) ^{2} \frac{1}{s e c ^{2} ( t )}}

(\frac{1}{csc ( t )})^{2} = \frac{1}{csc ^{2} ( t )}

(\frac{1}{csc ( t )})^{2}

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

:فعّل قانون القوى

= \frac{1 ^{2}}{csc ^{2} ( t )}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{csc ^{2} ( t )}

= \frac{( \frac{1}{s e c ( t )} ) ^{2} + ( \frac{1}{c s c ( t )} ) ^{2}}{\frac{1}{s e c ^{2} ( t )} \cdot \frac{1}{c s c ^{2} ( t )}}

(\frac{1}{sec ( t )})^{2} = \frac{1}{sec ^{2} ( t )}

(\frac{1}{sec ( t )})^{2}

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

:فعّل قانون القوى

= \frac{1 ^{2}}{sec ^{2} ( t )}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{sec ^{2} ( t )}

(\frac{1}{csc ( t )})^{2} = \frac{1}{csc ^{2} ( t )}

(\frac{1}{csc ( t )})^{2}

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

:فعّل قانون القوى

= \frac{1 ^{2}}{csc ^{2} ( t )}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{csc ^{2} ( t )}

= \frac{\frac{1}{s e c ^{2} ( t )} + \frac{1}{c s c ^{2} ( t )}}{\frac{1}{s e c ^{2} ( t )} \cdot \frac{1}{c s c ^{2} ( t )}}

\frac{1}{sec ^{2} ( t )} \cdot \frac{1}{csc ^{2} ( t )}

اضرب بـ:

\frac{1}{sec ^{2} ( t ) csc ^{2} ( t )}

\frac{1}{sec ^{2} ( t )} \cdot \frac{1}{csc ^{2} ( t )}

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

:اضرب كسور

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

1 \cdot 1 = 1 :

اضرب الأعداد

= \frac{1}{sec ^{2} ( t ) csc ^{2} ( t )}

= \frac{\frac{1}{s e c ^{2} ( t )} + \frac{1}{c s c ^{2} ( t )}}{\frac{1}{s e c ^{2} ( t ) c s c ^{2} ( t )}}

\frac{1}{sec ^{2} ( t )} + \frac{1}{csc ^{2} ( t )}

وحّد:

\frac{csc ^{2} ( t ) + sec ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )}

\frac{1}{sec ^{2} ( t )} + \frac{1}{csc ^{2} ( t )}

sec^{2} (t), csc^{2} (t)

المضاعف المشترك الأصغر لـ:

sec^{2} (t) csc^{2} (t)

sec^{2} (t), csc^{2} (t)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sec^{2} (t)

csc^{2} (t)

= sec^{2} (t) csc^{2} (t)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sec^{2} (t) csc^{2} (t)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{1}{sec ^{2} ( t )} :

multiply the denominator and numerator by

csc^{2} (t)

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

For

\frac{1}{csc ^{2} ( t )} :

multiply the denominator and numerator by

sec^{2} (t)

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

= \frac{csc ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )} + \frac{sec ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{csc ^{2} ( t ) + sec ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )}

= \frac{\frac{c s c ^{2} ( t ) + s e c ^{2} ( t )}{s e c ^{2} ( t ) c s c ^{2} ( t )}}{\frac{1}{s e c ^{2} ( t ) c s c ^{2} ( t )}}

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

:اقسم الكسور

= \frac{( csc ^{2} ( t ) + sec ^{2} ( t ) ) sec ^{2} ( t ) csc ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t ) \cdot 1}

بسّط

= \frac{( csc ^{2} ( t ) + sec ^{2} ( t ) ) sec ^{2} ( t ) csc ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )}

sec^{2} (t) :

إلغ العوامل المشتركة

= \frac{( csc ^{2} ( t ) + sec ^{2} ( t ) ) csc ^{2} ( t )}{csc ^{2} ( t )}

csc^{2} (t) :

إلغ العوامل المشتركة

= csc^{2} (t) + sec^{2} (t)

csc^{2} (t) + sec^{2} (t)

csc^{2} (t) + sec^{2} (t)

= sec^{2} (t) + csc^{2} (t)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

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