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人気のある三角関数 >

arccos(x)+arccos(2x)=arccos(1/2)

解

arccos (x) + arccos (2 x) = arccos (\frac{1}{2})

解

x = \frac{1}{2}

解答ステップ

arccos (x) + arccos (2 x) = arccos (\frac{1}{2})

a = b \Rightarrow cos (a) = cos (b)

cos (arccos (x) + arccos (2 x)) = cos (arccos (\frac{1}{2}))

次の恒等を使用する:

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

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

次の恒等式を使用する:

cos (arccos (x)) = x

次の恒等式を使用する:

cos (arccos (x)) = x

次の恒等式を使用する:

sin (arccos (x)) = 1 - x^{2}

次の恒等式を使用する:

sin (arccos (x)) = 1 - x^{2}

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

解く

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

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

以下で両辺を乗じる:

2

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

簡素化

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

平方根を削除する

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

両辺から

4 x^{2}

を引く

4 x^{2} - 2 1 - x^{2} 1 - (2 x)^{2} - 4 x^{2} = 1 - 4 x^{2}

簡素化

- 2 1 - x^{2} 1 - (2 x)^{2} = 1 - 4 x^{2}

両辺を2乗する:

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4}

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

(- 2 1 - x^{2} 1 - (2 x)^{2})^{2} = (1 - 4 x^{2})^{2}

拡張

(- 2 1 - x^{2} 1 - (2 x)^{2})^{2} : 4 - 20 x^{2} + 16 x^{4}

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

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

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

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

指数の規則を適用する:

(a \cdot b)^{n} = a^{n} b^{n}

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

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

累乗根の規則を適用する:

a = a^{\frac{1}{2}}

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

指数の規則を適用する:

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

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

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分数を乗じる:

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

= \frac{1 \cdot 2}{2}

共通因数を約分する:

2

= 1

= 1 - x^{2}

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

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

累乗根の規則を適用する:

a = a^{\frac{1}{2}}

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

指数の規則を適用する:

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

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

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分数を乗じる:

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

= \frac{1 \cdot 2}{2}

共通因数を約分する:

2

= 1

= 1 - (2 x)^{2}

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

2^{2} = 4

= 4 (1 - x^{2}) (1 - (2 x)^{2})

拡張

4 (1 - x^{2}) (1 - (2 x)^{2}) : 4 - 20 x^{2} + 16 x^{4}

4 (1 - x^{2}) (1 - (2 x)^{2})

指数の規則を適用する:

(a \cdot b)^{n} = a^{n} b^{n}

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

2^{2} = 4

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

拡張

(1 - x^{2}) (1 - 4 x^{2}) : 1 - 5 x^{2} + 4 x^{4}

(1 - x^{2}) (1 - 4 x^{2})

FOIL メソッドを適用する:

(a + b) (c + d) = a c + a d + b c + b d

a = 1, b = - x^{2}, c = 1, d = - 4 x^{2}

= 1 \cdot 1 + 1 \cdot (- 4 x^{2}) + (- x^{2}) \cdot 1 + (- x^{2}) (- 4 x^{2})

マイナス・プラスの規則を適用する

+ (- a) = - a, (- a) (- b) = ab

= 1 \cdot 1 - 1 \cdot 4 x^{2} - 1 \cdot x^{2} + 4 x^{2} x^{2}

簡素化

1 \cdot 1 - 1 \cdot 4 x^{2} - 1 \cdot x^{2} + 4 x^{2} x^{2} : 1 - 5 x^{2} + 4 x^{4}

1 \cdot 1 - 1 \cdot 4 x^{2} - 1 \cdot x^{2} + 4 x^{2} x^{2}

1 \cdot 1 = 1

1 \cdot 1

数を乗じる:

1 \cdot 1 = 1

= 1

1 \cdot 4 x^{2} = 4 x^{2}

1 \cdot 4 x^{2}

数を乗じる:

1 \cdot 4 = 4

= 4 x^{2}

1 \cdot x^{2} = x^{2}

1 \cdot x^{2}

乗算:

1 \cdot x^{2} = x^{2}

= x^{2}

4 x^{2} x^{2} = 4 x^{4}

4 x^{2} x^{2}

指数の規則を適用する:

a^{b} \cdot a^{c} = a^{b + c}

x^{2} x^{2} = x^{2 + 2}

= 4 x^{2 + 2}

数を足す:

2 + 2 = 4

= 4 x^{4}

= 1 - 4 x^{2} - x^{2} + 4 x^{4}

類似した元を足す:

- 4 x^{2} - x^{2} = - 5 x^{2}

= 1 - 5 x^{2} + 4 x^{4}

= 1 - 5 x^{2} + 4 x^{4}

= 4 (1 - 5 x^{2} + 4 x^{4})

拡張

4 (1 - 5 x^{2} + 4 x^{4}) : 4 - 20 x^{2} + 16 x^{4}

4 (1 - 5 x^{2} + 4 x^{4})

括弧を分配する

= 4 \cdot 1 + 4 (- 5 x^{2}) + 4 \cdot 4 x^{4}

マイナス・プラスの規則を適用する

+ (- a) = - a

= 4 \cdot 1 - 4 \cdot 5 x^{2} + 4 \cdot 4 x^{4}

簡素化

4 \cdot 1 - 4 \cdot 5 x^{2} + 4 \cdot 4 x^{4} : 4 - 20 x^{2} + 16 x^{4}

4 \cdot 1 - 4 \cdot 5 x^{2} + 4 \cdot 4 x^{4}

数を乗じる:

4 \cdot 1 = 4

= 4 - 4 \cdot 5 x^{2} + 4 \cdot 4 x^{4}

数を乗じる:

4 \cdot 5 = 20

= 4 - 20 x^{2} + 4 \cdot 4 x^{4}

数を乗じる:

4 \cdot 4 = 16

= 4 - 20 x^{2} + 16 x^{4}

= 4 - 20 x^{2} + 16 x^{4}

= 4 - 20 x^{2} + 16 x^{4}

= 4 - 20 x^{2} + 16 x^{4}

拡張

(1 - 4 x^{2})^{2} : 1 - 8 x^{2} + 16 x^{4}

(1 - 4 x^{2})^{2}

完全平方式を適用する:

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

a = 1, b = 4 x^{2}

= 1^{2} - 2 \cdot 1 \cdot 4 x^{2} + (4 x^{2})^{2}

簡素化

1^{2} - 2 \cdot 1 \cdot 4 x^{2} + (4 x^{2})^{2} : 1 - 8 x^{2} + 16 x^{4}

1^{2} - 2 \cdot 1 \cdot 4 x^{2} + (4 x^{2})^{2}

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 2 \cdot 1 \cdot 4 x^{2} + (4 x^{2})^{2}

2 \cdot 1 \cdot 4 x^{2} = 8 x^{2}

2 \cdot 1 \cdot 4 x^{2}

数を乗じる:

2 \cdot 1 \cdot 4 = 8

= 8 x^{2}

(4 x^{2})^{2} = 16 x^{4}

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

指数の規則を適用する:

(a \cdot b)^{n} = a^{n} b^{n}

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

(x^{2})^{2} : x^{4}

指数の規則を適用する:

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

= x^{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= x^{4}

= 4^{2} x^{4}

4^{2} = 16

= 16 x^{4}

= 1 - 8 x^{2} + 16 x^{4}

= 1 - 8 x^{2} + 16 x^{4}

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4}

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4}

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4}

解く

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4} : x = \frac{1}{2}, x = - \frac{1}{2}

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4}

4

を右側に移動します

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4}

両辺から

4

を引く

4 - 20 x^{2} + 16 x^{4} - 4 = 1 - 8 x^{2} + 16 x^{4} - 4

簡素化

- 20 x^{2} + 16 x^{4} = 16 x^{4} - 8 x^{2} - 3

- 20 x^{2} + 16 x^{4} = 16 x^{4} - 8 x^{2} - 3

8 x^{2}

を左側に移動します

- 20 x^{2} + 16 x^{4} = 16 x^{4} - 8 x^{2} - 3

両辺に

8 x^{2}

を足す

- 20 x^{2} + 16 x^{4} + 8 x^{2} = 16 x^{4} - 8 x^{2} - 3 + 8 x^{2}

簡素化

16 x^{4} - 12 x^{2} = 16 x^{4} - 3

16 x^{4} - 12 x^{2} = 16 x^{4} - 3

16 x^{4}

を左側に移動します

16 x^{4} - 12 x^{2} = 16 x^{4} - 3

両辺から

16 x^{4}

を引く

16 x^{4} - 12 x^{2} - 16 x^{4} = 16 x^{4} - 3 - 16 x^{4}

簡素化

- 12 x^{2} = - 3

- 12 x^{2} = - 3

以下で両辺を割る

- 12

- 12 x^{2} = - 3

以下で両辺を割る

- 12

\frac{- 12 x ^{2}}{- 12} = \frac{- 3}{- 12}

簡素化

\frac{- 12 x ^{2}}{- 12} = \frac{- 3}{- 12}

簡素化

\frac{- 12 x ^{2}}{- 12} : x^{2}

\frac{- 12 x ^{2}}{- 12}

分数の規則を適用する:

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

= \frac{12 x ^{2}}{12}

数を割る:

\frac{12}{12} = 1

= x^{2}

簡素化

\frac{- 3}{- 12} : \frac{1}{4}

\frac{- 3}{- 12}

分数の規則を適用する:

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

= \frac{3}{12}

共通因数を約分する:

3

= \frac{1}{4}

x^{2} = \frac{1}{4}

x^{2} = \frac{1}{4}

x^{2} = \frac{1}{4}

x^{2} = f (a)

の場合, 解は

x = f (a), - f (a)

x = \frac{1}{4}, x = - \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

累乗根の規則を適用する:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{1}{4}

累乗根の規則を適用する:

1 = 1

1 = 1

= \frac{1}{4}

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

a^{2} = a, a \geq 0

2^{2} = 2

= 2

= \frac{1}{2}

- \frac{1}{4} = - \frac{1}{2}

- \frac{1}{4}

累乗根の規則を適用する:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{1}{4}

累乗根の規則を適用する:

1 = 1

1 = 1

= - \frac{1}{4}

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

a^{2} = a, a \geq 0

2^{2} = 2

= 2

= - \frac{1}{2}

x = \frac{1}{2}, x = - \frac{1}{2}

x = \frac{1}{2}, x = - \frac{1}{2}

解を検算する:

x = \frac{1}{2}

真

, x = - \frac{1}{2}

真

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

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。

挿入

x = \frac{1}{2} :

真

(\frac{1}{2}) \cdot 2 (\frac{1}{2}) - 1 - (\frac{1}{2})^{2} 1 - (2 (\frac{1}{2}))^{2} = \frac{1}{2}

(\frac{1}{2}) \cdot 2 (\frac{1}{2}) - 1 - (\frac{1}{2})^{2} 1 - (2 (\frac{1}{2}))^{2} = \frac{1}{2}

(\frac{1}{2}) \cdot 2 (\frac{1}{2}) - 1 - (\frac{1}{2})^{2} 1 - (2 (\frac{1}{2}))^{2}

括弧を削除する:

(a) = a

= \frac{1}{2} \cdot 2 \cdot \frac{1}{2} - 1 - (\frac{1}{2})^{2} 1 - (2 \cdot \frac{1}{2})^{2}

\frac{1}{2} \cdot 2 \cdot \frac{1}{2} = \frac{1}{2}

\frac{1}{2} \cdot 2 \cdot \frac{1}{2}

分数を乗じる:

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

= \frac{1 \cdot 1 \cdot 2}{2 \cdot 2}

共通因数を約分する:

2

= \frac{1 \cdot 1}{2}

数を乗じる:

1 \cdot 1 = 1

= \frac{1}{2}

1 - (\frac{1}{2})^{2} 1 - (2 \cdot \frac{1}{2})^{2} = 0

1 - (\frac{1}{2})^{2} 1 - (2 \cdot \frac{1}{2})^{2}

1 - (\frac{1}{2})^{2} = \frac{3}{2}

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

(\frac{1}{2})^{2} = \frac{1}{4}

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

指数の規則を適用する:

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

= \frac{1 ^{2}}{2 ^{2}}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{2 ^{2}}

2^{2} = 4

= \frac{1}{4}

= 1 - \frac{1}{4}

結合

1 - \frac{1}{4} : \frac{3}{4}

1 - \frac{1}{4}

元を分数に変換する:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{1}{4}

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{1 \cdot 4 - 1}{4}

1 \cdot 4 - 1 = 3

1 \cdot 4 - 1

数を乗じる:

1 \cdot 4 = 4

= 4 - 1

数を引く:

4 - 1 = 3

= 3

= \frac{3}{4}

= \frac{3}{4}

累乗根の規則を適用する:

n \frac{a}{b} = \frac{n a}{n b},

, 以下を想定

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

= \frac{3}{2} - (2 \cdot \frac{1}{2})^{2} + 1

1 - (2 \cdot \frac{1}{2})^{2} = 0

1 - (2 \cdot \frac{1}{2})^{2}

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

(2 \cdot \frac{1}{2})^{2}

乗じる

2 \cdot \frac{1}{2} : 1

2 \cdot \frac{1}{2}

分数を乗じる:

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

= \frac{1 \cdot 2}{2}

共通因数を約分する:

2

= 1

= 1^{2}

規則を適用

1^{a} = 1

= 1

= 1 - 1

数を引く:

1 - 1 = 0

= 0

規則を適用

0 = 0

= 0

= 0 \cdot \frac{3}{2}

規則を適用

0 \cdot a = 0

= 0

= \frac{1}{2} - 0

\frac{1}{2} - 0 = \frac{1}{2}

= \frac{1}{2}

\frac{1}{2} = \frac{1}{2}

真

挿入

x = - \frac{1}{2} :

真

(- \frac{1}{2}) \cdot 2 (- \frac{1}{2}) - 1 - (- \frac{1}{2})^{2} 1 - (2 (- \frac{1}{2}))^{2} = \frac{1}{2}

(- \frac{1}{2}) \cdot 2 (- \frac{1}{2}) - 1 - (- \frac{1}{2})^{2} 1 - (2 (- \frac{1}{2}))^{2} = \frac{1}{2}

(- \frac{1}{2}) \cdot 2 (- \frac{1}{2}) - 1 - (- \frac{1}{2})^{2} 1 - (2 (- \frac{1}{2}))^{2}

括弧を削除する:

(- a) = - a, - (- a) = a

= \frac{1}{2} \cdot 2 \cdot \frac{1}{2} - 1 - (- \frac{1}{2})^{2} 1 - (- 2 \cdot \frac{1}{2})^{2}

\frac{1}{2} \cdot 2 \cdot \frac{1}{2} = \frac{1}{2}

\frac{1}{2} \cdot 2 \cdot \frac{1}{2}

分数を乗じる:

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

= \frac{1 \cdot 1 \cdot 2}{2 \cdot 2}

共通因数を約分する:

2

= \frac{1 \cdot 1}{2}

数を乗じる:

1 \cdot 1 = 1

= \frac{1}{2}

1 - (- \frac{1}{2})^{2} 1 - (- 2 \cdot \frac{1}{2})^{2} = 0

1 - (- \frac{1}{2})^{2} 1 - (- 2 \cdot \frac{1}{2})^{2}

1 - (- \frac{1}{2})^{2} = \frac{3}{2}

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

(- \frac{1}{2})^{2} = \frac{1}{4}

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

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

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

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

指数の規則を適用する:

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

= \frac{1 ^{2}}{2 ^{2}}

規則を適用

1^{a} = 1

1^{2} = 1

= \frac{1}{2 ^{2}}

2^{2} = 4

= \frac{1}{4}

= 1 - \frac{1}{4}

結合

1 - \frac{1}{4} : \frac{3}{4}

1 - \frac{1}{4}

元を分数に変換する:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{1}{4}

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{1 \cdot 4 - 1}{4}

1 \cdot 4 - 1 = 3

1 \cdot 4 - 1

数を乗じる:

1 \cdot 4 = 4

= 4 - 1

数を引く:

4 - 1 = 3

= 3

= \frac{3}{4}

= \frac{3}{4}

累乗根の規則を適用する:

n \frac{a}{b} = \frac{n a}{n b},

, 以下を想定

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

= \frac{3}{2} - (- 2 \cdot \frac{1}{2})^{2} + 1

1 - (- 2 \cdot \frac{1}{2})^{2} = 0

1 - (- 2 \cdot \frac{1}{2})^{2}

(- 2 \cdot \frac{1}{2})^{2} = 1

(- 2 \cdot \frac{1}{2})^{2}

乗じる

- 2 \cdot \frac{1}{2} : - 1

- 2 \cdot \frac{1}{2}

分数を乗じる:

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

= - \frac{1 \cdot 2}{2}

共通因数を約分する:

2

= - 1

= (- 1)^{2}

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

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

= 1^{2}

規則を適用

1^{a} = 1

= 1

= 1 - 1

数を引く:

1 - 1 = 0

= 0

規則を適用

0 = 0

= 0

= 0 \cdot \frac{3}{2}

規則を適用

0 \cdot a = 0

= 0

= \frac{1}{2} - 0

\frac{1}{2} - 0 = \frac{1}{2}

= \frac{1}{2}

\frac{1}{2} = \frac{1}{2}

真

解答は

x = \frac{1}{2}, x = - \frac{1}{2}

x = \frac{1}{2}, x = - \frac{1}{2}

元のequationに当てはめて解を検算する

arccos (x) + arccos (2 x) = arccos (\frac{1}{2})

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{1}{2} :

真

\frac{1}{2}

挿入

n = 1

\frac{1}{2}

arccos (x) + arccos (2 x) = arccos (\frac{1}{2})

の挿入向け

x = \frac{1}{2}

arccos (\frac{1}{2}) + arccos (2 \cdot \frac{1}{2}) = arccos (\frac{1}{2})

改良

1.04719 \dots = 1.04719 \dots

\Rightarrow 真

解答を確認する

- \frac{1}{2} :

偽

- \frac{1}{2}

挿入

n = 1

- \frac{1}{2}

arccos (x) + arccos (2 x) = arccos (\frac{1}{2})

の挿入向け

x = - \frac{1}{2}

arccos (- \frac{1}{2}) + arccos (2 (- \frac{1}{2})) = arccos (\frac{1}{2})

改良

5.23598 \dots = 1.04719 \dots

\Rightarrow 偽

x = \frac{1}{2}

グラフ

人気の例

tan(θ)=(2sqrt(3))/3 sin(θ),0<= θ<= 2pi

tan (θ) = \frac{2 3}{3} sin (θ), 0 \leq θ \leq 2 π

solvefor x,cos(x+y)=-cos(x)

so l v e f or x, cos (x + y) = - cos (x)

4+2tan^2(x)=5tan^2(x)+3

4 + 2 tan^{2} (x) = 5 tan^{2} (x) + 3

solvefor θ,sec(θ)=2

so l v e f or θ, sec (θ) = 2

2 cos (4 θ) = 0