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受欢迎的三角函数 >

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

解答

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

解答

x \in R 无解

求解步骤

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

两边减去

sec (x)

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

化简

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

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

将项转换为分式:

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

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

因为分母相等，所以合并分式:

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

= \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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

化简

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

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

乘开

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

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

使用分配律:

a (b + c) = ab + a c

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

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

使用加减运算法则

+ (- a) = - a

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

化简

- 1 \cdot sec (x) - sec (x) sec (x) : - sec (x) - sec^{2} (x)

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

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

1 \cdot sec (x)

乘以:

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

= sec (x)

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

sec (x) sec (x)

使用指数法则:

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

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

= sec^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sec^{2} (x)

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

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

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

同类项相加:

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

= - sec (x)

= - sec (x)

- sec (x) = 0

两边除以

- 1

- sec (x) = 0

两边除以

- 1

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

化简

sec (x) = 0

sec (x) = 0

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

x \in R 无解

作图

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