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prove csc(2A)+cot(2A)=cot(A)

الحلّ

prove

csc (2 A) + cot (2 A) = cot (A)

صحيح

خطوات الحلّ

csc (2 A) + cot (2 A) = cot (A)

قم بالعمل على الطرف الأيسر

csc (2 A) + cot (2 A)

sin, cos :

عبّر بواسطة

cot (2 A) + csc (2 A)

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

:Use the basic trigonometric identity

= \frac{cos ( 2 A )}{sin ( 2 A )} + csc (2 A)

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

:Use the basic trigonometric identity

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

\frac{cos ( 2 A )}{sin ( 2 A )} + \frac{1}{sin ( 2 A )}

بسّط:

\frac{cos ( 2 A ) + 1}{sin ( 2 A )}

\frac{cos ( 2 A )}{sin ( 2 A )} + \frac{1}{sin ( 2 A )}

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

فعّل القانون

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

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

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

Rewrite using trig identities

\frac{1 + cos ( 2 A )}{sin ( 2 A )}

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

:فعّل متطابقة الزاوية المضاعفة

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

cos (2 x) = 2 cos^{2} (x) - 1

:فعّل متطابقة الزاوية المضاعفة

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

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

بسّط:

\frac{cos ( A )}{sin ( A )}

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

1 + 2 cos^{2} (A) - 1 = 2 cos^{2} (A)

1 + 2 cos^{2} (A) - 1

جمّع التعابير المتشابهة

= 2 cos^{2} (A) + 1 - 1

1 - 1 = 0

= 2 cos^{2} (A)

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

\frac{2}{2} = 1 :

اقسم الأعداد

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

cos (A) :

إلغ العوامل المشتركة

= \frac{cos ( A )}{sin ( A )}

= \frac{cos ( A )}{sin ( A )}

= \frac{cos ( A )}{sin ( A )}

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

:Use the basic trigonometric identity

= cot (A)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e sin (x) - cos (x) = 1 - sin (2 x)

p ro v e 9 sec^{2} (x) - 5 tan^{2} (x) = 5 + 4 sec^{2} (x)

p ro v e sin (3 A) = 3 sin (A) - 4 sin^{3} (A)

p ro v e cot^{4} (x) + cot^{2} (x) = cot^{2} (x) csc^{2} (x)

p ro v e cot^{3} (x) = cot (x) (csc^{2} (x) - 1)