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受欢迎的三角函数 >

证明 sec(x)(sec(x)-cos(x))=tan^2(x)

解答

证明

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

解答

真

求解步骤

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

调整左侧

sec (x) (sec (x) - cos (x))

乘开

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

(- cos (x) + sec (x)) sec (x)

= sec (x) (- cos (x) + sec (x))

使用分配律:

a (b + c) = ab + a c

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

= sec (x) (- cos (x)) + sec (x) sec (x)

使用加减运算法则

+ (- a) = - a

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

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

使用三角恒等式改写

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

sec (x) cos (x) = 1

sec (x) cos (x)

用 sin, cos 表示

sec (x) cos (x)

使用基本三角恒等式:

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

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

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

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

分式相乘:

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

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

约分:

cos (x)

= 1

= 1

= sec^{2} (x) - 1

使用毕达哥拉斯恒等式:

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

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

= tan^{2} (x)

= tan^{2} (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

证明 1/(sin(x))-sin(x)=cot(x)cos(x)

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

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

证明 (sin(θ)-1)/(cos(θ))=tan(θ)-sec(θ)

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

证明 cot(x)*sec(x)=csc(x)

p ro v e cot (x) \cdot sec (x) = csc (x)

证明 2-csc^2(θ)=1-cot^2(θ)

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