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受欢迎的三角函数 >

tan(45+x)+tan(45-x)=4

解答

tan (4 5^{\circ} + x) + tan (4 5^{\circ} - x) = 4

解答

x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

+1

弧度

x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

求解步骤

tan (4 5^{\circ} + x) + tan (4 5^{\circ} - x) = 4

使用三角恒等式改写

tan (4 5^{\circ} + x) + tan (4 5^{\circ} - x) = 4

使用三角恒等式改写

tan (4 5^{\circ} + x)

使用基本三角恒等式:

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

= \frac{sin ( 4 5 ^{\circ} + x )}{cos ( 4 5 ^{\circ} + x )}

使用角和恒等式:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= \frac{sin ( 4 5 ^{\circ} ) cos ( x ) + cos ( 4 5 ^{\circ} ) sin ( x )}{cos ( 4 5 ^{\circ} + x )}

使用角和恒等式:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= \frac{sin ( 4 5 ^{\circ} ) cos ( x ) + cos ( 4 5 ^{\circ} ) sin ( x )}{cos ( 4 5 ^{\circ} ) cos ( x ) - sin ( 4 5 ^{\circ} ) sin ( x )}

化简

\frac{sin ( 4 5 ^{\circ} ) cos ( x ) + cos ( 4 5 ^{\circ} ) sin ( x )}{cos ( 4 5 ^{\circ} ) cos ( x ) - sin ( 4 5 ^{\circ} ) sin ( x )} : \frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )}

\frac{sin ( 4 5 ^{\circ} ) cos ( x ) + cos ( 4 5 ^{\circ} ) sin ( x )}{cos ( 4 5 ^{\circ} ) cos ( x ) - sin ( 4 5 ^{\circ} ) sin ( x )}

sin (4 5^{\circ}) cos (x) + cos (4 5^{\circ}) sin (x) = \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

sin (4 5^{\circ}) cos (x) + cos (4 5^{\circ}) sin (x)

化简

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

使用以下普通恒等式:

sin (4 5^{\circ}) = \frac{2}{2}

sin (x)

周期表（周期为

36 0^{\circ} n

"）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{2}{2} cos (x) + cos (4 5^{\circ}) sin (x)

化简

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

使用以下普通恒等式:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

周期表（周期为

36 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

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

= \frac{\frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{cos ( 4 5 ^{\circ} ) cos ( x ) - sin ( 4 5 ^{\circ} ) sin ( x )}

cos (4 5^{\circ}) cos (x) - sin (4 5^{\circ}) sin (x) = \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

cos (4 5^{\circ}) cos (x) - sin (4 5^{\circ}) sin (x)

化简

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

使用以下普通恒等式:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

周期表（周期为

36 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{2}{2} cos (x) - sin (4 5^{\circ}) sin (x)

化简

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

使用以下普通恒等式:

sin (4 5^{\circ}) = \frac{2}{2}

sin (x)

周期表（周期为

36 0^{\circ} n

"）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

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

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

乘

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

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

分式相乘:

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

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

= \frac{\frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} - \frac{2}{2} sin ( x )}

乘

\frac{2}{2} sin (x) : \frac{2 sin ( x )}{2}

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

分式相乘:

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

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

= \frac{\frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

乘

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

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

分式相乘:

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

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

= \frac{\frac{2 c o s ( x )}{2} + \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

乘

\frac{2}{2} sin (x) : \frac{2 sin ( x )}{2}

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

分式相乘:

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

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

= \frac{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

合并分式

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

使用法则

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

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

= \frac{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}

合并分式

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

使用法则

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

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

= \frac{\frac{2 c o s ( x ) + 2 s i n ( x )}{2}}{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}

分式相除:

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

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

约分:

2

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

因式分解出通项

2

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

因式分解出通项

2

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

约分:

2

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

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

使用基本三角恒等式:

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

= \frac{sin ( 4 5 ^{\circ} - x )}{cos ( 4 5 ^{\circ} - x )}

使用角差恒等式:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= \frac{sin ( 4 5 ^{\circ} ) cos ( x ) - cos ( 4 5 ^{\circ} ) sin ( x )}{cos ( 4 5 ^{\circ} - x )}

使用角差恒等式:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

= \frac{sin ( 4 5 ^{\circ} ) cos ( x ) - cos ( 4 5 ^{\circ} ) sin ( x )}{cos ( 4 5 ^{\circ} ) cos ( x ) + sin ( 4 5 ^{\circ} ) sin ( x )}

化简

\frac{sin ( 4 5 ^{\circ} ) cos ( x ) - cos ( 4 5 ^{\circ} ) sin ( x )}{cos ( 4 5 ^{\circ} ) cos ( x ) + sin ( 4 5 ^{\circ} ) sin ( x )} : \frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )}

\frac{sin ( 4 5 ^{\circ} ) cos ( x ) - cos ( 4 5 ^{\circ} ) sin ( x )}{cos ( 4 5 ^{\circ} ) cos ( x ) + sin ( 4 5 ^{\circ} ) sin ( x )}

sin (4 5^{\circ}) cos (x) - cos (4 5^{\circ}) sin (x) = \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

sin (4 5^{\circ}) cos (x) - cos (4 5^{\circ}) sin (x)

化简

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

使用以下普通恒等式:

sin (4 5^{\circ}) = \frac{2}{2}

sin (x)

周期表（周期为

36 0^{\circ} n

"）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{2}{2} cos (x) - cos (4 5^{\circ}) sin (x)

化简

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

使用以下普通恒等式:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

周期表（周期为

36 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

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

= \frac{\frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{cos ( 4 5 ^{\circ} ) cos ( x ) + sin ( 4 5 ^{\circ} ) sin ( x )}

cos (4 5^{\circ}) cos (x) + sin (4 5^{\circ}) sin (x) = \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

cos (4 5^{\circ}) cos (x) + sin (4 5^{\circ}) sin (x)

化简

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

使用以下普通恒等式:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

周期表（周期为

36 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{2}{2} cos (x) + sin (4 5^{\circ}) sin (x)

化简

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

使用以下普通恒等式:

sin (4 5^{\circ}) = \frac{2}{2}

sin (x)

周期表（周期为

36 0^{\circ} n

"）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

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

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

乘

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

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

分式相乘:

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

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

= \frac{\frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} + \frac{2}{2} sin ( x )}

乘

\frac{2}{2} sin (x) : \frac{2 sin ( x )}{2}

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

分式相乘:

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

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

= \frac{\frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

乘

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

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

分式相乘:

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

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

= \frac{\frac{2 c o s ( x )}{2} - \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

乘

\frac{2}{2} sin (x) : \frac{2 sin ( x )}{2}

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

分式相乘:

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

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

= \frac{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

合并分式

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

使用法则

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

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

= \frac{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}{\frac{2 c o s ( x ) + 2 s i n ( x )}{2}}

合并分式

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

使用法则

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

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

= \frac{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}{\frac{2 c o s ( x ) + 2 s i n ( x )}{2}}

分式相除:

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

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

约分:

2

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

因式分解出通项

2

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

因式分解出通项

2

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

约分:

2

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

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

\frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )} + \frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )} = 4

化简

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

\frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )} + \frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )}

cos (x) - sin (x), cos (x) + sin (x)

的最小公倍数:

(cos (x) - sin (x)) (cos (x) + sin (x))

cos (x) - sin (x), cos (x) + sin (x)

最小公倍数 (LCM)

计算出由出现在

cos (x) - sin (x)

或

cos (x) + sin (x)

中的因子组成的表达式

= (cos (x) - sin (x)) (cos (x) + sin (x))

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

(cos (x) - sin (x)) (cos (x) + sin (x))

对于

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

将分母和分子乘以

cos (x) + sin (x)

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

对于

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

将分母和分子乘以

cos (x) - sin (x)

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

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

因为分母相等，所以合并分式:

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

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

乘开

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

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

(cos (x) + sin (x))^{2} : cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x)

使用完全平方公式:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = cos (x), b = sin (x)

= cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x)

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

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

使用完全平方公式:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = cos (x), b = sin (x)

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

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

化简

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

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

同类项相加:

2 cos (x) sin (x) - 2 cos (x) sin (x) = 0

= cos^{2} (x) + sin^{2} (x) + cos^{2} (x) + sin^{2} (x)

同类项相加:

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

= 2 cos^{2} (x) + sin^{2} (x) + sin^{2} (x)

同类项相加:

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

= 2 cos^{2} (x) + 2 sin^{2} (x)

= 2 cos^{2} (x) + 2 sin^{2} (x)

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

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

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

两边减去

4

\frac{2 cos ^{2} ( x ) + 2 sin ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} - 4 = 0

化简

\frac{2 cos ^{2} ( x ) + 2 sin ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} - 4 : \frac{- 2 cos ^{2} ( x ) + 6 sin ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

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

将项转换为分式:

4 = \frac{4 ( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

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

因为分母相等，所以合并分式:

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

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

乘开

2 cos^{2} (x) + 2 sin^{2} (x) - 4 (cos (x) - sin (x)) (cos (x) + sin (x)) : - 2 cos^{2} (x) + 6 sin^{2} (x)

2 cos^{2} (x) + 2 sin^{2} (x) - 4 (cos (x) - sin (x)) (cos (x) + sin (x))

乘开

- 4 (cos (x) - sin (x)) (cos (x) + sin (x)) : - 4 cos^{2} (x) + 4 sin^{2} (x)

乘开

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

(cos (x) - sin (x)) (cos (x) + sin (x))

使用平方差公式:

(a - b) (a + b) = a^{2} - b^{2}

a = cos (x), b = sin (x)

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

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

乘开

- 4 (cos^{2} (x) - sin^{2} (x)) : - 4 cos^{2} (x) + 4 sin^{2} (x)

- 4 (cos^{2} (x) - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = - 4, b = cos^{2} (x), c = sin^{2} (x)

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

使用加减运算法则

- (- a) = a

= - 4 cos^{2} (x) + 4 sin^{2} (x)

= - 4 cos^{2} (x) + 4 sin^{2} (x)

= 2 cos^{2} (x) + 2 sin^{2} (x) - 4 cos^{2} (x) + 4 sin^{2} (x)

化简

2 cos^{2} (x) + 2 sin^{2} (x) - 4 cos^{2} (x) + 4 sin^{2} (x) : - 2 cos^{2} (x) + 6 sin^{2} (x)

2 cos^{2} (x) + 2 sin^{2} (x) - 4 cos^{2} (x) + 4 sin^{2} (x)

同类项相加:

2 cos^{2} (x) - 4 cos^{2} (x) = - 2 cos^{2} (x)

= - 2 cos^{2} (x) + 2 sin^{2} (x) + 4 sin^{2} (x)

同类项相加:

2 sin^{2} (x) + 4 sin^{2} (x) = 6 sin^{2} (x)

= - 2 cos^{2} (x) + 6 sin^{2} (x)

= - 2 cos^{2} (x) + 6 sin^{2} (x)

= \frac{- 2 cos ^{2} ( x ) + 6 sin ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

\frac{- 2 cos ^{2} ( x ) + 6 sin ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 2 cos^{2} (x) + 6 sin^{2} (x) = 0

分解

- 2 cos^{2} (x) + 6 sin^{2} (x) : 2 (3 sin (x) + cos (x)) (3 sin (x) - cos (x))

- 2 cos^{2} (x) + 6 sin^{2} (x)

将

6

改写为

3 \cdot 2

= - 2 cos^{2} (x) + 3 \cdot 2 sin^{2} (x)

因式分解出通项

2

= 2 (- cos^{2} (x) + 3 sin^{2} (x))

分解

3 sin^{2} (x) - cos^{2} (x) : (3 sin (x) + cos (x)) (3 sin (x) - cos (x))

3 sin^{2} (x) - cos^{2} (x)

将

3 sin^{2} (x) - cos^{2} (x)

改写为

(3 sin (x))^{2} - cos^{2} (x)

3 sin^{2} (x) - cos^{2} (x)

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

= (3)^{2} sin^{2} (x) - cos^{2} (x)

使用指数法则:

a^{m} b^{m} = (ab)^{m}

(3)^{2} sin^{2} (x) = (3 sin (x))^{2}

= (3 sin (x))^{2} - cos^{2} (x)

= (3 sin (x))^{2} - cos^{2} (x)

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

(3 sin (x))^{2} - cos^{2} (x) = (3 sin (x) + cos (x)) (3 sin (x) - cos (x))

= (3 sin (x) + cos (x)) (3 sin (x) - cos (x))

= 2 (3 sin (x) + cos (x)) (3 sin (x) - cos (x))

2 (3 sin (x) + cos (x)) (3 sin (x) - cos (x)) = 0

分别求解每个部分

3 sin (x) + cos (x) = 0 or 3 sin (x) - cos (x) = 0

3 sin (x) + cos (x) = 0 : x = 15 0^{\circ} + 18 0^{\circ} n

3 sin (x) + cos (x) = 0

使用三角恒等式改写

3 sin (x) + cos (x) = 0

在两边除以

cos (x), cos (x) \neq = 0

\frac{3 sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{3 sin ( x )}{cos ( x )} + 1 = 0

使用基本三角恒等式:

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

3 tan (x) + 1 = 0

3 tan (x) + 1 = 0

将

1

到右边

3 tan (x) + 1 = 0

两边减去

1

3 tan (x) + 1 - 1 = 0 - 1

化简

3 tan (x) = - 1

3 tan (x) = - 1

两边除以

3

3 tan (x) = - 1

两边除以

3

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

化简

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

化简

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

约分:

3

= tan (x)

化简

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{3}

- \frac{1}{3}

有理化:

- \frac{3}{3}

- \frac{1}{3}

乘以共轭根式

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

的通解

tan (x)

周期表（周期为

18 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 15 0^{\circ} + 18 0^{\circ} n

x = 15 0^{\circ} + 18 0^{\circ} n

3 sin (x) - cos (x) = 0 : x = 3 0^{\circ} + 18 0^{\circ} n

3 sin (x) - cos (x) = 0

使用三角恒等式改写

3 sin (x) - cos (x) = 0

在两边除以

cos (x), cos (x) \neq = 0

\frac{3 sin ( x ) - cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{3 sin ( x )}{cos ( x )} - 1 = 0

使用基本三角恒等式:

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

3 tan (x) - 1 = 0

3 tan (x) - 1 = 0

将

1

到右边

3 tan (x) - 1 = 0

两边加上

1

3 tan (x) - 1 + 1 = 0 + 1

化简

3 tan (x) = 1

3 tan (x) = 1

两边除以

3

3 tan (x) = 1

两边除以

3

\frac{3 tan ( x )}{3} = \frac{1}{3}

化简

\frac{3 tan ( x )}{3} = \frac{1}{3}

化简

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

约分:

3

= tan (x)

化简

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

乘以共轭根式

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

的通解

tan (x)

周期表（周期为

18 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 3 0^{\circ} + 18 0^{\circ} n

x = 3 0^{\circ} + 18 0^{\circ} n

合并所有解

x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

作图

流行的例子

1-cos(θ)=sin(θ)

1 - cos (θ) = sin (θ)

sqrt(3)cot((3pi)/7 x)+3=0

3 cot (\frac{3 π}{7} x) + 3 = 0

sin (u) = 1

2sqrt(2)cos(x)+3=5

22 cos (x) + 3 = 5

4tan^2(x)+14tan(x)+12=0

4 tan^{2} (x) + 14 tan (x) + 12 = 0