证明 tan(2x)(1-tan^2(x))=2tan(x)

解答

证明

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

真

求解步骤

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

调整左侧

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

使用三角恒等式改写

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

使用倍角公式:

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

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

化简

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

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

分式相乘:

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

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

约分:

1 - tan^{2} (x)

= 2 tan (x)

= 2 tan (x)

= 2 tan (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

p ro v e csc (x) - cos (x) \cdot cot (x) = sin (x)

p ro v e sin (x) \cdot csc (x) = 1

p ro v e csc^{2} (θ) (1 - cos^{2} (θ)) = sin (648 0^{\circ})

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

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