English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

prove (2-sec^2(x))/(sec^2(x))=cos(2x)

الحلّ

prove

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

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

\frac{2 - sec ^{2} ( x )}{sec ^{2} ( x )}

sin, cos :

عبّر بواسطة

\frac{2 - sec ^{2} ( x )}{sec ^{2} ( x )}

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

:Use the basic trigonometric identity

= \frac{2 - ( \frac{1}{c o s ( x )} ) ^{2}}{( \frac{1}{c o s ( x )} ) ^{2}}

\frac{2 - ( \frac{1}{c o s ( x )} ) ^{2}}{( \frac{1}{c o s ( x )} ) ^{2}}

بسّط:

2 cos^{2} (x) - 1

\frac{2 - ( \frac{1}{c o s ( x )} ) ^{2}}{( \frac{1}{c o s ( x )} ) ^{2}}

(\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{2 - ( \frac{1}{c o s ( x )} ) ^{2}}{\frac{1}{c o s ^{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{2 - \frac{1}{c o s ^{2} ( x )}}{\frac{1}{c o s ^{2} ( x )}}

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

: استخدم ميزات الكسور التالية

= \frac{( 2 - \frac{1}{c o s ^{2} ( x )} ) cos ^{2} ( x )}{1}

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

= \frac{\frac{2 c o s ^{2} ( x ) - 1}{c o s ^{2} ( x )} cos ^{2} ( x )}{1}

\frac{a}{1} = a

: استخدم ميزات الكسور التالية

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

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

:اضرب كسور

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

cos^{2} (x) :

إلغ العوامل المشتركة

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

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

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

Rewrite using trig identities

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

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

:فعّل متطابقة الزاوية المضاعفة

= cos (2 x)

= cos (2 x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e \frac{cos ( β )}{1 - sin ( β )} = sec (β) + tan (β)

p ro v e tan^{2} (y) - sin^{2} (y) = tan^{2} (y) sin^{2} (y)

p ro v e sin (θ) (1 + cot^{2} (θ)) = csc (θ)

p ro v e 5 + cot^{2} (x) = 4 + csc^{2} (x)

p ro v e sin^{2} (2 x) = \frac{1 - cos ( 4 x )}{2}