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العربية

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

Решение

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

x = 2 πn

Градусы

x = 0^{\circ} + 36 0^{\circ} n

Шаги решения

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

Вычтите

sec (x)

с обеих сторон

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

Перепишите используя тригонометрические тождества

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

Испльзуйте основное тригонометрическое тождество:

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

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

Упростите

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

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

Примените правило возведения в степень:

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

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

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

Расширить

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

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

Примените распределительный закон:

a (b + c) = ab + a c

a = cos^{2} (x), b = 1, c = \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

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

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

Упростить

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

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

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

1 \cdot cos^{2} (x)

Умножьте:

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

= cos^{2} (x)

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

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

Умножьте дроби:

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

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

Отмените общий множитель:

cos^{2} (x)

= sin^{2} (x)

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

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

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

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

Используйте основное тригонометрическое тождество (тождество Пифагора):

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

= - sec (x) + 1

- sec (x) + 1 = 0

Переместите

1

вправо

- sec (x) + 1 = 0

Вычтите

1

с обеих сторон

- sec (x) + 1 - 1 = 0 - 1

После упрощения получаем

- sec (x) = - 1

- sec (x) = - 1

Разделите обе стороны на

- 1

- sec (x) = - 1

Разделите обе стороны на

- 1

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

После упрощения получаем

sec (x) = 1

sec (x) = 1

Общие решения для

sec (x) = 1

sec (x)

таблица периодичности с циклом

2 πn

x = 0 + 2 πn

x = 0 + 2 πn

Решить

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn