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受欢迎的三角函数 >

证明 (1+cos(2x))/(2sin^2(x))=cot^2(x)

解答

证明

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

解答

真

求解步骤

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

调整左侧

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

使用三角恒等式改写

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

使用倍角公式:

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

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

化简

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

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

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

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

对同类项分组

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

1 - 1 = 0

= 2 cos^{2} (x)

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

数字相除:

\frac{2}{2} = 1

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

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

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

使用三角恒等式改写

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

使用基本三角恒等式:

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

\frac{cos ( x ) cot ( x )}{sin ( x )}

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

使用基本三角恒等式:

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

cot (x) cot (x)

化简

cot (x) cot (x) : cot^{2} (x)

cot (x) cot (x)

使用指数法则:

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

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

= cot^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cot^{2} (x)

cot^{2} (x)

cot^{2} (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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