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prove sec^2(x)+csc^2(x)=(sec(x)csc(x))^2

الحلّ

prove

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

الحلّ

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

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

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

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

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

:فعّل قانون القوى

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

(\frac{1}{cos ( x )})^{2}

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

:فعّل قانون القوى

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

sin^{2} (x), cos^{2} (x)

المضاعف المشترك الأصغر لـ:

sin^{2} (x) cos^{2} (x)

sin^{2} (x), cos^{2} (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin^{2} (x)

cos^{2} (x)

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

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin^{2} (x) cos^{2} (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{1}{sin ^{2} ( x )} :

multiply the denominator and numerator by

cos^{2} (x)

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

For

\frac{1}{cos ^{2} ( x )} :

multiply the denominator and numerator by

sin^{2} (x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

قم بالعمل على الطرف الأيمن

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

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

(\frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )})^{2}

بسّط:

\frac{1}{sin ^{2} ( x ) cos ^{2} ( x )}

(\frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )})^{2}

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

اضرب بـ:

\frac{1}{sin ( x ) cos ( x )}

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

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

:اضرب كسور

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

1 \cdot 1 = 1 :

اضرب الأعداد

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

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

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

:فعّل قانون القوى

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

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

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

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

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

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

أمثلة شائعة

prove tan(x)=(sin(x)csc(x))/(cot(x))prove

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prove (1+sin(x))/(sin(x))=1+csc(x)prove

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

prove cos(2y)=(1-tan^2(y))/(1+tan^2(y))prove

cos (2 y) = \frac{1 - tan ^{2} ( y )}{1 + tan ^{2} ( y )}

prove sin(θ)cot^2(θ)=cos^2(θ)csc(θ)prove

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