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受欢迎的三角函数 >

证明 (sec(x)+tan(x))(1-sin(x))=cos(x)

解答

证明

(sec (x) + tan (x)) (1 - sin (x)) = cos (x)

解答

真

求解步骤

(sec (x) + tan (x)) (1 - sin (x)) = cos (x)

调整左侧

(sec (x) + tan (x)) (1 - sin (x))

用 sin, cos 表示

(1 - sin (x)) (sec (x) + tan (x))

使用基本三角恒等式:

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

= (1 - sin (x)) (\frac{1}{cos ( x )} + tan (x))

使用基本三角恒等式:

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

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

化简

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

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

化简

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

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

使用法则

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

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

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

分式相乘:

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

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

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

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

使用三角恒等式改写

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

乘开

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

(1 + sin (x)) (1 - sin (x))

使用平方差公式:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = sin (x)

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

使用法则

1^{a} = 1

1^{2} = 1

= 1 - sin^{2} (x)

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

使用毕达哥拉斯恒等式:

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

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

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

约分:

cos (x)

= cos (x)

= cos (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 tan(A)+cot(A)=sec(A)csc(A)

p ro v e tan (A) + cot (A) = sec (A) csc (A)

证明 (cos(x))/(sec(x)-1)-(cos(x))/(sec(x)+1)=(2cos(x))/(tan^2(x))

p ro v e \frac{cos ( x )}{sec ( x ) - 1} - \frac{cos ( x )}{sec ( x ) + 1} = \frac{2 cos ( x )}{tan ^{2} ( x )}

证明 (cot(a))/(cos(a))=csc(a)

p ro v e \frac{cot ( a )}{cos ( a )} = csc (a)

证明 cos(2θ)=(2-sec^2(θ))/(sec^2(θ))

p ro v e cos (2 θ) = \frac{2 - sec ^{2} ( θ )}{sec ^{2} ( θ )}

证明 (tan(x)+sec(x))(1-sin(x))=cos(x)

p ro v e (tan (x) + sec (x)) (1 - sin (x)) = cos (x)