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人気のある三角関数 >

sec^2(x)-1= 1/(cot(x))

解

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

解

x = \frac{π}{4} + πn

+1

度

x = 4 5^{\circ} + 18 0^{\circ} n

解答ステップ

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

両辺を2乗する

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

両辺から

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

を引く

(sec^{2} (x) - 1)^{2} - \frac{1}{cot ^{2} ( x )} = 0

簡素化

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

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

元を分数に変換する:

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

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

\frac{cot ^{2} ( x ) ( sec ^{2} ( x ) - 1 ) ^{2} - 1}{cot ^{2} ( x )} = 0

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

cot^{2} (x) (sec^{2} (x) - 1)^{2} - 1 = 0

三角関数の公式を使用して書き換える

- 1 + (- 1 + sec^{2} (x))^{2} cot^{2} (x)

ピタゴラスの公式を使用する:

sec^{2} (x) = tan^{2} (x) + 1

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

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

(tan^{2} (x))^{2} = tan^{4} (x)

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

指数の規則を適用する:

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

= tan^{2 \cdot 2} (x)

数を乗じる:

2 \cdot 2 = 4

= tan^{4} (x)

= - 1 + tan^{4} (x) cot^{2} (x)

- 1 + cot^{2} (x) tan^{4} (x) = 0

因数

- 1 + cot^{2} (x) tan^{4} (x) : (tan^{2} (x) cot (x) + 1) (tan^{2} (x) cot (x) - 1)

- 1 + cot^{2} (x) tan^{4} (x)

- 1 + cot^{2} (x) tan^{4} (x)

を書き換え

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

- 1 + cot^{2} (x) tan^{4} (x)

指数の規則を適用する:

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

tan^{4} (x) = (tan^{2} (x))^{2}

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

指数の規則を適用する:

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

cot^{2} (x) (tan^{2} (x))^{2} = (cot (x) tan^{2} (x))^{2}

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

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

2乗の差の公式を適用する:

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

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

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

(tan^{2} (x) cot (x) + 1) (tan^{2} (x) cot (x) - 1) = 0

各部分を別個に解く

tan^{2} (x) cot (x) + 1 = 0 or tan^{2} (x) cot (x) - 1 = 0

tan^{2} (x) cot (x) + 1 = 0 : x = \frac{3 π}{4} + πn

tan^{2} (x) cot (x) + 1 = 0

三角関数の公式を使用して書き換える

1 + cot (x) tan^{2} (x)

基本的な三角関数の公式を使用する:

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

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

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

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

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

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

指数の規則を適用する:

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

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

規則を適用

1^{a} = 1

1^{2} = 1

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

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

分数を乗じる:

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

= \frac{1 \cdot cot ( x )}{cot ^{2} ( x )}

乗算:

1 \cdot cot (x) = cot (x)

= \frac{cot ( x )}{cot ^{2} ( x )}

共通因数を約分する:

cot (x)

= \frac{1}{cot ( x )}

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

1 + \frac{1}{cot ( x )} = 0

以下で両辺を乗じる:

cot (x)

1 + \frac{1}{cot ( x )} = 0

以下で両辺を乗じる:

cot (x)

1 \cdot cot (x) + \frac{1}{cot ( x )} cot (x) = 0 \cdot cot (x)

簡素化

1 \cdot cot (x) + \frac{1}{cot ( x )} cot (x) = 0 \cdot cot (x)

簡素化

1 \cdot cot (x) : cot (x)

1 \cdot cot (x)

乗算:

1 \cdot cot (x) = cot (x)

= cot (x)

簡素化

\frac{1}{cot ( x )} cot (x) : 1

\frac{1}{cot ( x )} cot (x)

分数を乗じる:

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

= \frac{1 \cdot cot ( x )}{cot ( x )}

共通因数を約分する:

cot (x)

= 1

簡素化

0 \cdot cot (x) : 0

0 \cdot cot (x)

規則を適用

0 \cdot a = 0

= 0

cot (x) + 1 = 0

cot (x) + 1 = 0

cot (x) + 1 = 0

1

を右側に移動します

cot (x) + 1 = 0

両辺から

1

を引く

cot (x) + 1 - 1 = 0 - 1

簡素化

cot (x) = - 1

cot (x) = - 1

解を検算する

未定義の (特異) 点を求める:

cot (x) = 0

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

の分母をゼロに比較する

cot (x) = 0

以下の点は定義されていない

cot (x) = 0

未定義のポイントを解に組み合わせる:

cot (x) = - 1

以下の一般解

cot (x) = - 1

cot (x)

πn

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

tan^{2} (x) cot (x) - 1 = 0 : x = \frac{π}{4} + πn

tan^{2} (x) cot (x) - 1 = 0

三角関数の公式を使用して書き換える

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

基本的な三角関数の公式を使用する:

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

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

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

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

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

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

指数の規則を適用する:

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

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

規則を適用

1^{a} = 1

1^{2} = 1

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

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

分数を乗じる:

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

= \frac{1 \cdot cot ( x )}{cot ^{2} ( x )}

乗算:

1 \cdot cot (x) = cot (x)

= \frac{cot ( x )}{cot ^{2} ( x )}

共通因数を約分する:

cot (x)

= \frac{1}{cot ( x )}

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

- 1 + \frac{1}{cot ( x )} = 0

以下で両辺を乗じる:

cot (x)

- 1 + \frac{1}{cot ( x )} = 0

以下で両辺を乗じる:

cot (x)

- 1 \cdot cot (x) + \frac{1}{cot ( x )} cot (x) = 0 \cdot cot (x)

簡素化

- 1 \cdot cot (x) + \frac{1}{cot ( x )} cot (x) = 0 \cdot cot (x)

簡素化

- 1 \cdot cot (x) : - cot (x)

- 1 \cdot cot (x)

乗算:

1 \cdot cot (x) = cot (x)

= - cot (x)

簡素化

\frac{1}{cot ( x )} cot (x) : 1

\frac{1}{cot ( x )} cot (x)

分数を乗じる:

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

= \frac{1 \cdot cot ( x )}{cot ( x )}

共通因数を約分する:

cot (x)

= 1

簡素化

0 \cdot cot (x) : 0

0 \cdot cot (x)

規則を適用

0 \cdot a = 0

= 0

- cot (x) + 1 = 0

- cot (x) + 1 = 0

- cot (x) + 1 = 0

1

を右側に移動します

- cot (x) + 1 = 0

両辺から

1

を引く

- cot (x) + 1 - 1 = 0 - 1

簡素化

- cot (x) = - 1

- cot (x) = - 1

以下で両辺を割る

- 1

- cot (x) = - 1

以下で両辺を割る

- 1

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

簡素化

cot (x) = 1

cot (x) = 1

解を検算する

未定義の (特異) 点を求める:

cot (x) = 0

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

の分母をゼロに比較する

cot (x) = 0

以下の点は定義されていない

cot (x) = 0

未定義のポイントを解に組み合わせる:

cot (x) = 1

以下の一般解

cot (x) = 1

cot (x)

πn

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

すべての解を組み合わせる

x = \frac{3 π}{4} + πn, x = \frac{π}{4} + πn

元のequationに当てはめて解を検算する

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

に当てはめて解を確認する

equationに一致しないものを削除する。

解答を確認する

\frac{3 π}{4} + πn :

偽

\frac{3 π}{4} + πn

挿入

n = 1

\frac{3 π}{4} + π 1

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

の挿入向け

x = \frac{3 π}{4} + π 1

sec^{2} (\frac{3 π}{4} + π 1) - 1 = \frac{1}{cot ( \frac{3 π}{4} + π 1 )}

改良

1 = - 1

\Rightarrow 偽

解答を確認する

\frac{π}{4} + πn :

真

\frac{π}{4} + πn

挿入

n = 1

\frac{π}{4} + π 1

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

の挿入向け

x = \frac{π}{4} + π 1

sec^{2} (\frac{π}{4} + π 1) - 1 = \frac{1}{cot ( \frac{π}{4} + π 1 )}

改良

1 = 1

\Rightarrow 真

x = \frac{π}{4} + πn

グラフ

人気の例

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2tan^4(x)-tan^2(x)-15=0

2 tan^{4} (x) - tan^{2} (x) - 15 = 0

cos(x)=-sqrt(1/2)

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

6cos(θ)=-6(1+cos(θ))

6 cos (θ) = - 6 (1 + cos (θ))

2sin^2(x)+5sin(x)+3=0

2 sin^{2} (x) + 5 sin (x) + 3 = 0