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人気のある三角関数 >

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

解

証明する

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

解

真

解答ステップ

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

右側を操作する

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

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

= sin^{2} (x) + sin^{2} (x) (sec^{2} (x) - 1)

簡素化

sin^{2} (x) + sin^{2} (x) (sec^{2} (x) - 1) : sin^{2} (x) sec^{2} (x)

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

拡張

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

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

分配法則を適用する:

a (b - c) = ab - a c

a = sin^{2} (x), b = sec^{2} (x), c = 1

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

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

乗算:

1 \cdot sin^{2} (x) = sin^{2} (x)

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

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

類似した元を足す:

sin^{2} (x) - sin^{2} (x) = 0

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

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

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

サイン, コサインで表わす

sec^{2} (x) sin^{2} (x)

基本的な三角関数の公式を使用する:

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

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

簡素化

(\frac{1}{cos ( x )})^{2} sin^{2} (x) : \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

(\frac{1}{cos ( x )})^{2} sin^{2} (x)

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

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

指数の規則を適用する:

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

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

規則を適用

1^{a} = 1

1^{2} = 1

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

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

分数を乗じる:

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

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

乗算:

1 \cdot sin^{2} (x) = sin^{2} (x)

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

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

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

三角関数の公式を使用して書き換える

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

基本的な三角関数の公式を使用する:

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

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

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

基本的な三角関数の公式を使用する:

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

tan (x) tan (x)

簡素化

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

tan (x) tan (x)

指数の規則を適用する:

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

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

= tan^{1 + 1} (x)

数を足す:

1 + 1 = 2

= tan^{2} (x)

tan^{2} (x)

tan^{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-sin(x)sec(x)=-tan(x)

p ro v e - sin (x) sec (x) = - tan (x)

証明する cos(4x)-cos(3x)cos(x)=sin(2x)-4cos(2x)sin(2x)

p ro v e cos (4 x) - cos (3 x) cos (x) = sin (2 x) - 4 cos (2 x) sin (2 x)

証明する cos(4x)=1-8sin^2(x)cos(x)

p ro v e cos (4 x) = 1 - 8 sin^{2} (x) cos (x)

証明する (sec^2(x)cot(x))+cot(x)=tan(x)

p ro v e (sec^{2} (x) cot (x)) + cot (x) = tan (x)

証明する 6cos^2(A)-5=1-6sin^2(A)

p ro v e 6 cos^{2} (A) - 5 = 1 - 6 sin^{2} (A)