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

الحلّ

prove

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

الحلّ

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

2 tan (x) sec (x) csc (x)

sin, cos :

عبّر بواسطة

2 csc (x) sec (x) tan (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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 sin ( x ) \cdot 2}{sin ( x ) cos ( x ) cos ( x )}

sin (x) :

إلغ العوامل المشتركة

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

1 \cdot 1 \cdot 2 = 2 :

اضرب الأعداد

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

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

cos (x) cos (x)

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

:فعّل قانون القوى

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

= cos^{1 + 1} (x)

1 + 1 = 2 :

اجمع الأعداد

= cos^{2} (x)

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

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

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

قم بالعمل على الطرف الأيمن

2 + 2 tan^{2} (x)

2 + 2 tan^{2} (x)

حلل إلى عوامل:

2 (1 + tan^{2} (x))

2 + 2 tan^{2} (x)

أعد الكتابة كـ

= 2 \cdot 1 + 2 tan^{2} (x)

2

قم باخراج العامل المشترك

= 2 (1 + tan^{2} (x))

= (1 + tan^{2} (x)) \cdot 2

sin, cos :

عبّر بواسطة

(1 + tan^{2} (x)) \cdot 2

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

:فعّل قانون القوى

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

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

اضرب

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

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

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

:اضرب كسور

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

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

بسّط

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

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

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

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