簡約 1-tan(x+pi/3)

解

簡約

1 - tan (x + \frac{π}{3})

1 - \frac{1 + 3 cot ( x )}{cot ( x ) - 3}

解答ステップ

1 - tan (x + \frac{π}{3})

三角関数の公式を使用して書き換える

= 1 - \frac{\frac{1}{2} sin ( x ) + \frac{3}{2} cos ( x )}{\frac{1}{2} cos ( x ) - \frac{3}{2} sin ( x )}

基本的な三角関数の公式を使用する:

sin (x) = \frac{1}{csc ( x )}

= \frac{cos ( x ) \frac{3}{2} + \frac{1}{2} \cdot \frac{1}{c s c ( x )}}{cos ( x ) \frac{1}{2} - \frac{3}{2} \cdot \frac{1}{c s c ( x )}}

簡素化

\frac{cos ( x ) \frac{3}{2} + \frac{1}{2} \cdot \frac{1}{c s c ( x )}}{cos ( x ) \frac{1}{2} - \frac{3}{2} \cdot \frac{1}{c s c ( x )}} : \frac{3 cos ( x ) csc ( x ) + 1}{cos ( x ) csc ( x ) - 3}

= \frac{3 cos ( x ) csc ( x ) + 1}{cos ( x ) csc ( x ) - 3}

csc (x) cos (x) = cot (x)

= 1 - \frac{3 cot ( x ) + 1}{cot ( x ) - 3}

\frac{1 + cot ( x ) 3}{cot ( x ) - 3} = \frac{1 + 3 cot ( x )}{cot ( x ) - 3}

= 1 - \frac{3 cot ( x ) + 1}{cot ( x ) - 3}