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شائع علم المثلثات >

(1+tan^2(x))/(1+sec(x))=sec(x)

الحلّ

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

الحلّ

x \in R لا يوجد حلّ لـ

خطوات الحلّ

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

من الطرفين

sec (x)

اطرح

\frac{1 + tan ^{2} ( x )}{1 + sec ( x )} - sec (x) = 0

\frac{1 + tan ^{2} ( x )}{1 + sec ( x )} - sec (x)

بسّط:

\frac{1 + tan ^{2} ( x ) - sec ( x ) ( 1 + sec ( x ) )}{1 + sec ( x )}

\frac{1 + tan ^{2} ( x )}{1 + sec ( x )} - sec (x)

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

\frac{1 + tan ^{2} ( x ) - sec ( x ) ( 1 + sec ( x ) )}{1 + sec ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

1 + tan^{2} (x) - sec (x) (1 + sec (x)) = 0

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

بسّط:

- sec (x)

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

- sec (x) (1 + sec (x))

وسٌع:

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

- sec (x) (1 + sec (x))

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = - sec (x), b = 1, c = sec (x)

= - sec (x) \cdot 1 + (- sec (x)) sec (x)

فعّل قوانين سالب-موجب

+ (- a) = - a

= - 1 \cdot sec (x) - sec (x) sec (x)

- 1 \cdot sec (x) - sec (x) sec (x)

بسّط:

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

- 1 \cdot sec (x) - sec (x) sec (x)

1 \cdot sec (x) = sec (x)

1 \cdot sec (x)

1 \cdot sec (x) = sec (x) :

اضرب

= sec (x)

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

sec (x) sec (x)

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

:فعّل قانون القوى

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

= sec^{1 + 1} (x)

1 + 1 = 2 :

اجمع الأعداد

= sec^{2} (x)

= - sec (x) - sec^{2} (x)

= - sec (x) - sec^{2} (x)

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

- sec^{2} (x) + sec^{2} (x) = 0 :

اجمع العناصر المتشابهة

= - sec (x)

= - sec (x)

- sec (x) = 0

- 1

اقسم الطرفين على

- sec (x) = 0

- 1

اقسم الطرفين على

\frac{- sec ( x )}{- 1} = \frac{0}{- 1}

بسّط

sec (x) = 0

sec (x) = 0

sec (x) \leq - 1 or sec (x) \geq 1

x \in R لا يوجد حلّ لـ

رسم

أمثلة شائعة

-sin^2(x)+2cos(x)-2=0

sin^4(x)+cos^2(x)=2