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受欢迎的三角函数 >

证明 2/(1+cos(2x))=sec^2(x)

解答

证明

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

解答

真

求解步骤

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

调整左侧

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

使用三角恒等式改写

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

使用倍角公式:

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

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

化简

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

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

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

数字相除:

\frac{2}{2} = 1

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

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

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

使用三角恒等式改写

使用基本三角恒等式:

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

\frac{1}{( \frac{1}{s e c ( x )} ) ^{2}}

化简

\frac{1}{( \frac{1}{s e c ( x )} ) ^{2}}

(\frac{1}{sec ( x )})^{2} = \frac{1}{sec ^{2} ( x )}

(\frac{1}{sec ( x )})^{2}

使用指数法则:

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

= \frac{1 ^{2}}{sec ^{2} ( x )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( x )}

= \frac{1}{\frac{1}{s e c ^{2} ( x )}}

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{sec ^{2} ( x )}{1}

使用法则

\frac{a}{1} = a

= sec^{2} (x)

sec^{2} (x)

sec^{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

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p ro v e 1 + tan^{2} (6 0^{\circ}) = sec^{2} (6 0^{\circ})

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