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受欢迎的三角函数 >

证明 sec^4(x)-tan^4(x)=1+2tan^2(x)

解答

证明

sec^{4} (x) - tan^{4} (x) = 1 + 2 tan^{2} (x)

解答

真

求解步骤

sec^{4} (x) - tan^{4} (x) = 1 + 2 tan^{2} (x)

调整左侧

sec^{4} (x) - tan^{4} (x)

分解

sec^{4} (x) - tan^{4} (x) : (sec^{2} (x) + tan^{2} (x)) (sec (x) + tan (x)) (sec (x) - tan (x))

sec^{4} (x) - tan^{4} (x)

将

sec^{4} (x) - tan^{4} (x)

改写为

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

sec^{4} (x) - tan^{4} (x)

使用指数法则:

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

sec^{4} (x) = (sec^{2} (x))^{2}

= (sec^{2} (x))^{2} - tan^{4} (x)

使用指数法则:

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

tan^{4} (x) = (tan^{2} (x))^{2}

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

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

使用平方差公式:

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

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

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

分解

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

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

使用平方差公式:

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

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

= (sec (x) + tan (x)) (sec (x) - tan (x))

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

化简

tan^{2} (x) + 1 + tan^{2} (x) : 2 tan^{2} (x) + 1

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

对同类项分组

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

同类项相加:

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

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

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

乘开

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

(sec (x) + tan (x)) (sec (x) - tan (x))

使用平方差公式:

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

a = sec (x), b = tan (x)

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

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

使用毕达哥拉斯恒等式:

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

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

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

化简

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

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

= 1 + 2 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 sin(-x)=-sin(x)

p ro v e sin (- x) = - sin (x)

证明 sin(x+pi)=-sin(x)

p ro v e sin (x + π) = - sin (x)

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p ro v e \frac{cos ( 2 X )}{sin ( X )} + \frac{sin ( 2 X )}{cos ( X )} = csc (X)

证明 tan^5(x)=tan^3(x)sec^2(x)-tan^3(x)

p ro v e tan^{5} (x) = tan^{3} (x) sec^{2} (x) - tan^{3} (x)

证明 (cos(θ))/(1-sin(θ))=sec(θ)+tan(θ)

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