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العربية

prove cot(2b)=(cot^2(b)-1)/(2cot(b))

الحلّ

prove

cot (2 b) = \frac{cot ^{2} ( b ) - 1}{2 cot ( b )}

صحيح

خطوات الحلّ

cot (2 b) = \frac{cot ^{2} ( b ) - 1}{2 cot ( b )}

قم بالعمل على الطرف الأيمن

\frac{cot ^{2} ( b ) - 1}{2 cot ( b )}

sin, cos :

عبّر بواسطة

\frac{- 1 + cot ^{2} ( b )}{2 cot ( b )}

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

:Use the basic trigonometric identity

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

\frac{- 1 + ( \frac{c o s ( b )}{s i n ( b )} ) ^{2}}{2 \cdot \frac{c o s ( b )}{s i n ( b )}}

بسّط:

\frac{- sin ^{2} ( b ) + cos ^{2} ( b )}{2 sin ( b ) cos ( b )}

\frac{- 1 + ( \frac{c o s ( b )}{s i n ( b )} ) ^{2}}{2 \cdot \frac{c o s ( b )}{s i n ( b )}}

2 \cdot \frac{cos ( b )}{sin ( b )}

اضرب بـ:

\frac{2 cos ( b )}{sin ( b )}

2 \cdot \frac{cos ( b )}{sin ( b )}

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

:اضرب كسور

= \frac{cos ( b ) \cdot 2}{sin ( b )}

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

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

:فعّل قانون القوى

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

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

: استخدم ميزات الكسور التالية

= \frac{( - 1 + \frac{c o s ^{2} ( b )}{s i n ^{2} ( b )} ) sin ( b )}{cos ( b ) \cdot 2}

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

وحّد:

\frac{- sin ^{2} ( b ) + cos ^{2} ( b )}{sin ^{2} ( b )}

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

1 \cdot sin^{2} (b) = sin^{2} (b) :

اضرب

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

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

\frac{- sin ^{2} ( b ) + cos ^{2} ( b )}{sin ^{2} ( b )} sin (b)

اضرب بـ:

\frac{- sin ^{2} ( b ) + cos ^{2} ( b )}{sin ( b )}

\frac{- sin ^{2} ( b ) + cos ^{2} ( b )}{sin ^{2} ( b )} sin (b)

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

:اضرب كسور

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

sin (b) :

إلغ العوامل المشتركة

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

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

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

: استخدم ميزات الكسور التالية

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

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

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

Rewrite using trig identities

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

2 sin (x) cos (x) = sin (2 x)

:فعّل متطابقة الزاوية المضاعفة

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

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

:فعّل متطابقة الزاوية المضاعفة

= \frac{cos ( 2 b )}{sin ( 2 b )}

= \frac{cos ( 2 b )}{sin ( 2 b )}

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

:Use the basic trigonometric identity

= cot (2 b)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e 1 + tan^{2} (a) = sec^{2} (a)

p ro v e 2 cos (x) cos (y) = cos (x + y) + cos (x - y)

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

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

p ro v e (1 + tan^{2} (u)) (1 - sin^{2} (u)) = 1