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受欢迎的三角函数 >

证明 sin^2(2x)=(1-cos(4x))/2

解答

证明

sin^{2} (2 x) = \frac{1 - cos ( 4 x )}{2}

解答

真

求解步骤

sin^{2} (2 x) = \frac{1 - cos ( 4 x )}{2}

调整右侧

\frac{1 - cos ( 4 x )}{2}

使用三角恒等式改写

\frac{1 - cos ( 4 x )}{2}

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

使用倍角公式:

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

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

化简

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

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

乘开

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

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

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

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

打开括号

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

使用加减运算法则

- (- a) = a, - (a) = - a

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

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

1 - 1 = 0

= 2 sin^{2} (2 x)

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

数字相除:

\frac{2}{2} = 1

= sin^{2} (2 x)

= sin^{2} (2 x)

= sin^{2} (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 cot(2θ)=(csc(θ)-2sin(θ))/(2cos(θ))

p ro v e cot (2 θ) = \frac{csc ( θ ) - 2 sin ( θ )}{2 cos ( θ )}

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p ro v e sin^{2} (θ) sec^{2} (θ) csc^{2} (θ) = sec^{2} (θ)

证明 (csc^2(x))/(cot(x))=sec(x)csc(x)

p ro v e \frac{csc ^{2} ( x )}{cot ( x )} = sec (x) csc (x)

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p ro v e \frac{cos ( A )}{1 + sin ( A )} + \frac{cos ( A )}{1 - sin ( A )} = 2 sec (A)

证明 cot(x/2)=(sin(x))/(1-cos(x))

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