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인기 있는 삼각법 >

증명 1/(1+tan^2(x))+1/(1+cot^2(x))=1

해법

증명

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

해법

참

솔루션 단계

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

왼쪽 조작

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

죄로 표현하라, 왜냐하면

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

기본 삼각형 항등식 사용:

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

= \frac{1}{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}} + \frac{1}{1 + tan ^{2} ( x )}

기본 삼각형 항등식 사용:

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

= \frac{1}{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}} + \frac{1}{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}

\frac{1}{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}} + \frac{1}{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}} = 1

\frac{1}{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}} + \frac{1}{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}

\frac{1}{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}} = \frac{sin ^{2} ( x )}{sin ^{2} ( x ) + cos ^{2} ( x )}

\frac{1}{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

지수 규칙 적용:

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

= \frac{1}{1 + \frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}

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

합류하다:

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

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

요소를 분수로 변환:

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

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

분모가 같기 때문에, 분수를 합친다:

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

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

곱하다:

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

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

= \frac{1}{\frac{s i n ^{2} ( x ) + c o s ^{2} ( x )}{s i n ^{2} ( x )}}

분수 규칙 적용:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

\frac{1}{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}} = \frac{cos ^{2} ( x )}{cos ^{2} ( x ) + sin ^{2} ( x )}

\frac{1}{1 + ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}

지수 규칙 적용:

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

= \frac{1}{1 + \frac{s i n ^{2} ( x )}{c o s ^{2} ( x )}}

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

합류하다:

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

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

요소를 분수로 변환:

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

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

분모가 같기 때문에, 분수를 합친다:

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

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

곱하다:

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

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

= \frac{1}{\frac{c o s ^{2} ( x ) + s i n ^{2} ( x )}{c o s ^{2} ( x )}}

분수 규칙 적용:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

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

규칙 적용

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

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

규칙 적용

\frac{a}{a} = 1

= 1

= 1

= 1

우리는 양측이 같은 형태를 취할 수 있다는 것을 보여주었습니다

\Rightarrow 참

인기 있는 예

sin (3 x) = 1

cos(2x)+cos(x)=0

cos (2 x) + cos (x) = 0

csch (ln (2))

tan(2x)=sqrt(3)

tan (2 x) = 3

solvefor x,tan(x)+sec(x)=2cos(x)을 위해 해결하다

x, tan (x) + sec (x) = 2 cos (x)