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人気のある三角関数 >

証明する tan(45+x)+tan(45-x)=2sec(2x)

解

証明する

tan (4 5^{\circ} + x) + tan (4 5^{\circ} - x) = 2 sec (2 x)

解

真

解答ステップ

tan (4 5^{\circ} + x) + tan (4 5^{\circ} - x) = 2 sec (2 x)

左側を操作する

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

三角関数の公式を使用して書き換える

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

基本的な三角関数の公式を使用する:

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

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

角の和の公式を使用する:

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 )}

角の和の公式を使用する:

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 )}

簡素化

\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)

簡素化

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

sin (4 5^{\circ})

次の自明恒等式を使用する:

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

sin (x)

36 0^{\circ} n

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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)

簡素化

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

cos (4 5^{\circ})

次の自明恒等式を使用する:

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

cos (x)

36 0^{\circ} n

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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)

簡素化

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

cos (4 5^{\circ})

次の自明恒等式を使用する:

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

cos (x)

36 0^{\circ} n

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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)

簡素化

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

sin (4 5^{\circ})

次の自明恒等式を使用する:

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

sin (x)

36 0^{\circ} n

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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 )}

乗じる

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

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

分数を乗じる:

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 )}

乗じる

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

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

分数を乗じる:

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}}

乗じる

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

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

分数を乗じる:

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}}

乗じる

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

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

分数を乗じる:

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}}

分数を組み合わせる

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

規則を適用

\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}}

分数を組み合わせる

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

規則を適用

\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}}

分数を割る:

\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 ) )}

共通因数を約分する:

2

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

共通項をくくり出す

2

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

共通項をくくり出す

2

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

共通因数を約分する:

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

三角関数の公式を使用して書き換える

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

基本的な三角関数の公式を使用する:

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

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

角の差の公式を使用する:

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 )}

角の差の公式を使用する:

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 )}

簡素化

\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)

簡素化

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

sin (4 5^{\circ})

次の自明恒等式を使用する:

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

sin (x)

36 0^{\circ} n

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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)

簡素化

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

cos (4 5^{\circ})

次の自明恒等式を使用する:

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

cos (x)

36 0^{\circ} n

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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)

簡素化

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

cos (4 5^{\circ})

次の自明恒等式を使用する:

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

cos (x)

36 0^{\circ} n

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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)

簡素化

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

sin (4 5^{\circ})

次の自明恒等式を使用する:

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

sin (x)

36 0^{\circ} n

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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 )}

乗じる

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

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

分数を乗じる:

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 )}

乗じる

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

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

分数を乗じる:

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}}

乗じる

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

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

分数を乗じる:

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}}

乗じる

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

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

分数を乗じる:

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}}

分数を組み合わせる

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

規則を適用

\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}}

分数を組み合わせる

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

規則を適用

\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}}

分数を割る:

\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 ) )}

共通因数を約分する:

2

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

共通項をくくり出す

2

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

共通項をくくり出す

2

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

共通因数を約分する:

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 )}

簡素化

\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 )}

以下の最小公倍数:

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

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

最小公倍数 (LCM)

cos (x) - sin (x)

または以下のいずれかに現れる因数で構成された式を計算する:

cos (x) + sin (x)

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

LCMに基づいて分数を調整する

該当する分母を乗じてLCMに変えるために

必要な量で各分子を乗じる

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

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

の場合

:

分母と分子に以下を乗じる:

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 )}{cos ( x ) + sin ( x )}

の場合

:

分母と分子に以下を乗じる:

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 ) )}

分母が等しいので, 分数を組み合わせる:

\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 ) )}

拡張

(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)

完全平方式を適用する:

(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)

完全平方式を適用する:

(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)

簡素化

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)

類似した元を足す:

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

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

類似した元を足す:

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

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

類似した元を足す:

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 ) )}

因数

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

因数

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

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

共通項をくくり出す

2

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

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

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

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

簡素化

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

拡張

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

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

2乗の差の公式を適用する:

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

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

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

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

2倍角の公式を使用:

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

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

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

三角関数の公式を使用して書き換える

基本的な三角関数の公式を使用する:

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

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

簡素化

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

分数の規則を適用する:

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

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

規則を適用

\frac{a}{1} = a

= 2 sec (2 x)

2 sec (2 x)

2 sec (2 x)

両辺を同じ形式にできることを証明した

\Rightarrow 真

人気の例

証明する cos^2(8x)-sin^2(8x)=cos(16x)

p ro v e cos^{2} (8 x) - sin^{2} (8 x) = cos (16 x)

証明する cos^2(2x)= 1/2+1/2 cos(4x)

p ro v e cos^{2} (2 x) = \frac{1}{2} + \frac{1}{2} cos (4 x)

証明する 2sin(-t)cos(-t)=2sin(-t)cos(t)

p ro v e 2 sin (- t) cos (- t) = 2 sin (- t) cos (t)

証明する sin^2(α)+sin^2(α)tan^2(α)=tan^2(α)

p ro v e sin^{2} (α) + sin^{2} (α) tan^{2} (α) = tan^{2} (α)

証明する cos(75)=sqrt((1+cos(150))/2)

p ro v e cos (7 5^{\circ}) = \frac{1 + cos ( 15 0 ^{\circ} )}{2}