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Problèmes Trigonométrie courants

prouver ((cot(x)))/((csc(x)))=cos(x)	`prove` (cot(`x`))(csc(`x`)) =cos(`x`)
prouver ((sec^2(t)))/(sec^2(t)-1)=csc^2(t)	`prove` (sec²(`t`))sec²(`t`)−1 =csc²(`t`)
prouver sin^2(x)=cos(2x)+2	`prove` sin²(`x`)=cos(2`x`)+2
prouver 1+cos^2(x)=2-sin^2(x)	`prove` 1+cos²(`x`)=2−sin²(`x`)
prouver sin^2(x)=cos(2x)-2	`prove` sin²(`x`)=cos(2`x`)−2
prouver (tan(θ)cot(θ))/(cos(θ))=sec(θ)	`prove` tan(θ)cot(θ)cos(θ) =sec(θ)
prouver 2(cos(θ-1))^2=cos^4(θ)-sin^4(θ)	`prove` 2(cos(θ−1))²=cos⁴(θ)−sin⁴(θ)
prouver cos(x)= 15/17	`prove` cos(`x`)=1517
prouver csc(A)-sin(A)=(cos(A))(cot(A))	`prove` csc(`A`)−sin(`A`)=(cos(`A`))(cot(`A`))
prouver (cos^2(x))/(cos^2(x))=1	`prove` cos²(`x`)cos²(`x`) =1
prouver (1-sec(x))/(csc(x))=cos(x)(cot(x))	`prove` 1−sec(`x`)csc(`x`) =cos(`x`)(cot(`x`))
prouver csc(x)tan(x)sec(x)=sec^2(x)	`prove` csc(`x`)tan(`x`)sec(`x`)=sec²(`x`)
prouver tan(x)sec^4(x)=(sin(x))/(cos^5(x))	`prove` tan(`x`)sec⁴(`x`)=sin(`x`)cos⁵(`x`)
prouver csc(x)+cot(x)sec(x)-1=tan(x)	`prove` csc(`x`)+cot(`x`)sec(`x`)−1=tan(`x`)
prouver (3csc(x)-3sin(x))/(tan(x)-cot(x))=3cos^3(x)	`prove` 3csc(`x`)−3sin(`x`)tan(`x`)−cot(`x`) =3cos³(`x`)
prouver 9cos(x)+6sin(x)=10	`prove` 9cos(`x`^◦)+6sin(`x`^◦)=10
prouver (tan^2(A))/(sec^2(A))=sin^2(A)	`prove` tan²(`A`)sec²(`A`) =sin²(`A`)
prouver 5cos^2(x)-2cos(x)-3-sin^2(x)=0	`prove` 5cos²(`x`)−2cos(`x`)−3−sin²(`x`)=0
prouver cos^4(a)+1-sin^4(a)=2cos^2(a)	`prove` cos⁴(`a`)+1−sin⁴(`a`)=2cos²(`a`)
prouver 3-4cos^2(x)=(2sin(x)+1)(2sin(x)-1)	`prove` 3−4cos²(`x`)=(2sin(`x`)+1)(2sin(`x`)−1)
prouver csc^2(θ)=(1/(sin(θ)))^2	`prove` csc²(θ)=(1sin(θ) )²
prouver (1+tan(x))/(1+1/(tan(x)))=tan(x)	`prove` 1+tan(`x`)1+1tan(`x`) =tan(`x`)
prouver cos(2θ)= 1/(sec(2θ))	`prove` cos(2θ)=1sec(2θ)
prouver (cos(3x)-cos(x))=-2sin(2x)sin(x)	`prove` (cos(3`x`)−cos(`x`))=−2sin(2`x`)sin(`x`)
prouver sec(pi/2-y)=csc(y)	`prove` sec(π2 −`y`)=csc(`y`)
prouver cos(2x-pi/2)=cos(pi/2-2x)	`prove` cos(2`x`−π2 )=cos(π2 −2`x`)
prouver sin(pi/2-x)cot(pi/2+x)=-sin(x)	`prove` sin(π2 −`x`)cot(π2 +`x`)=−sin(`x`)
prouver cos(x)csc(x)tan(x)=1	`prove` cos(`x`)· csc(`x`)· tan(`x`)=1
prouver cos^2(x) 1/(cos^2(x))=1	`prove` cos²(`x`)1cos²(`x`) =1
prouver (sin((4pi)/3))=-(sqrt(3))/2	`prove` (sin(4π3 ))=−√32
prouver sec(x)-sin^2(x)=cos(x)	`prove` sec(`x`)−sin²(`x`)=cos(`x`)
prouver cot^2(x)=csc^2(x)(1-sin^2(x))	`prove` cot²(`x`)=csc²(`x`)(1−sin²(`x`))
prouver (cos(2θ))/(-sin^2(θ))=cos^2(θ)	`prove` cos(2θ)−sin²(θ) =cos²(θ)
prouver sin(x)+cot(x)(cos(x))=csc(x)	`prove` sin(`x`)+cot(`x`)(cos(`x`))=csc(`x`)
prouver 1/(csc(x)-1)=(sin(x))/1	`prove` 1csc(`x`)−1 =sin(`x`)1
prouver sec(θ)cos(θ)csc(θ)=cot(θ)	`prove` sec(θ)cos(θ)csc(θ)=cot(θ)
prouver csc^2(x)(1-cos^2(x))=tan(420)	`prove` csc²(`x`)(1−cos²(`x`))=tan(420^◦)
prouver cos(θ+30)-sin(θ+60)=-sin(θ)	`prove` cos(θ+30^◦)−sin(θ+60^◦)=−sin(θ)
prouver tan(a)*cot(a)=sin^2(a)+cos^2(a)	`prove` tan(`a`)· cot(`a`)=sin²(`a`)+cos²(`a`)
prouver tan(x)+(cos(x))/(1-sin(x))=sec(x)	`prove` tan(`x`)+cos(`x`)1−sin(`x`) =sec(`x`)
prouver sin(x)cos(x)=tan(x)	`prove` sin(`x`)cos(`x`)=tan(`x`)
prouver cot((15pi)/8)=cot((7pi)/8)	`prove` cot(15π8 )=cot(7π8 )
prouver sin^4(x)=(sin^2(x))^2	`prove` sin⁴(`x`)=(sin²(`x`))²
prouver sin(2x)-cos(2x)= 1/2	`prove` sin(2`x`)−cos(2`x`)=12
prouver cos^{(2)}(θ)(1+tan^{(2)}(θ))=1	`prove` cos⁽²⁾(θ)(1+tan⁽²⁾(θ))=1
prouver 1-2sin^2(y)+sin^4(y)=cos^4(y)	`prove` 1−2sin²(`y`)+sin⁴(`y`)=cos⁴(`y`)
prouver (sin(x)+cos(x))^2-2sin(x)cos(x)=1	`prove` (sin(`x`)+cos(`x`))²−2sin(`x`)cos(`x`)=1
prouver (1-sin(3a))(sin(3a)+1)=cos^2(3a)	`prove` (1−sin(3`a`))(sin(3`a`)+1)=cos²(3`a`)
prouver (sin(x))/(1+cos(2x))=tan(x)	`prove` sin(`x`)1+cos(2`x`) =tan(`x`)
prouver sec(t)(csc(t)(tan(t)+cot(t)))=sec^2(t)+csc^2(t)	`prove` sec(`t`)(csc(`t`)(tan(`t`)+cot(`t`)))=sec²(`t`)+csc²(`t`)
prouver (1+sin(x))^2+cos^2(x)=2+2sin(x)	`prove` (1+sin(`x`))²+cos²(`x`)=2+2sin(`x`)
prouver cot(60)=(cos(60))/(sin(60))	`prove` cot(60^◦)=cos(60^◦)sin(60^◦)
prouver tan(-x)tan(pi/2-x)=-1	`prove` tan(−`x`)tan(π2 −`x`)=−1
prouver tan(pi-θ)=-tan(x)	`prove` tan(π−θ)=−tan(`x`)
prouver cot(θ)(sin(θ)+tan(θ))=cos(θ)+1	`prove` cot(θ)(sin(θ)+tan(θ))=cos(θ)+1
prouver (2-sin^2(x))csc^2(x)=cot^2(x)	`prove` (2−sin²(`x`))csc²(`x`)=cot²(`x`)
prouver 1/(tan(A))+tan(A)= 2/(sin(2A))	`prove` 1tan(`A`) +tan(`A`)=2sin(2`A`)
prouver 1+sin(θ)=cos(θ)	`prove` 1+sin(θ)=cos(θ)
prouver 1+((tan^2(x)))/(1+sec(x))=sec(x)	`prove` 1+(tan²(`x`))1+sec(`x`) =sec(`x`)
prouver csc^2(x)*cos^2(x)=cot^2(x)	`prove` csc²(`x`)· cos²(`x`)=cot²(`x`)
prouver 1/(sec^3(x)cos^4(x))=sec(x)	`prove` 1sec³(`x`)cos⁴(`x`) =sec(`x`)
prouver csc^2(θ)+1=cot^2(θ)	`prove` csc²(θ)+1=cot²(θ)
prouver 1+tan^2(B)=sec^2(B)	`prove` 1+tan²(`B`)=sec²(`B`)
prouver cos^2(7θ)-sin^2(7θ)=cos(14θ)	`prove` cos²(7θ)−sin²(7θ)=cos(14θ)
prouver sin^4(x)-(3/(4*sin^2(x)))+1=1	`prove` sin⁴(`x`)−(34· sin²(`x`) )+1=1
prouver arccot(x)=tan(x)	`prove` arccot(`x`)=tan(`x`)
prouver cot(θ)+tan(θ)=sec(θ)+csc(θ)	`prove` cot(θ)+tan(θ)=sec(θ)+csc(θ)
prouver 2sin(θ)+sin(2θ)=0	`prove` 2sin(θ)+sin(2θ)=0
prouver cos^2(x)+cos(x)-1+sin^2(x)=cos(x)	`prove` cos²(`x`)+cos(`x`)−1+sin²(`x`)=cos(`x`)
prouver (2sin(x)cos(x))/(cos(x))=2	`prove` 2sin(`x`)cos(`x`)cos(`x`) =2
prouver tan(x-(3pi)/2)=-cot(x)	`prove` tan(`x`−3π2 )=−cot(`x`)
prouver sin(θ)(cos^2(θ))/(sin(θ))=csc(θ)	`prove` sin(θ)cos²(θ)sin(θ) =csc(θ)
prouver (tan^2(a)+1)/(sec(a))=sec(a)	`prove` tan²(`a`)+1sec(`a`) =sec(`a`)
prouver 1/(tan(β)+cot(β))=sin(β)cos(β)	`prove` 1tan(β)+cot(β) =sin(β)cos(β)
prouver cos(300)=1-2sin^2(150)	`prove` cos(300^◦)=1−2sin²(150^◦)
prouver csc^2(x)+3cot^2(x)-5=4(cot(x)-1)	`prove` csc²(`x`)+3cot²(`x`)−5=4(cot(`x`)−1)
prouver (3)((cos(2z))^2)/2 =(3cos(4z))/4	`prove` (3)(cos(2`z`))²2 =3cos(4`z`)4
prouver-2sin^2(x)+cos(x)+1=0	`prove` −2sin²(`x`)+cos(`x`)+1=0
prouver (2cot(u))/(csc^2(u)-2)=tan(2u)	`prove` 2cot(`u`)csc²(`u`)−2 =tan(2`u`)
prouver csc(2x)+cot(2x)=(1+cos(2x))/(sin(2x))	`prove` csc(2`x`)+cot(2`x`)=1+cos(2`x`)sin(2`x`)
prouver 1-2sin^2(t)=2cos^2(t)-1	`prove` 1−2sin²(`t`)=2cos²(`t`)−1
prouver 1/(cos^2(θ))=sec^2(θ)	`prove` 1cos²(θ) =sec²(θ)
prouver-cos(2t)sin(2t)+sin(2t)cos(2t)+0=0	`prove` −cos(2`t`)sin(2`t`)+sin(2`t`)cos(2`t`)+0=0
prouver (tan(θ)sin(θ))/(sec(θ)-1)=1+cos(θ)	`prove` tan(θ)sin(θ)sec(θ)−1 =1+cos(θ)
prouver 1-2sin^2(x)=-1+cos^2(x)	`prove` 1−2sin²(`x`)=−1+cos²(`x`)
prouver (sin(x)sin(x))/(cos(x))=cos(x)	`prove` sin(`x`)sin(`x`)cos(`x`) =cos(`x`)
prouver (cos(x))/5 = 1/5*cos(x)	`prove` cos(`x`)5 =15 · cos(`x`)
prouver (1+tan(x))/(sec(x))=cos(x)+sin(x)	`prove` 1+tan(`x`)sec(`x`) =cos(`x`)+sin(`x`)
prouver (sin(4x))/4 =(sin(x)cos(x))/2	`prove` sin(4`x`)4 =sin(`x`)cos(`x`)2
prouver (sin(x)tan(x))/(cos(x)+1)=sec(x)-1	`prove` sin(`x`)tan(`x`)cos(`x`)+1 =sec(`x`)−1
prouver sin(a+b)-sin(a-b)=2sin(a)sin(b)	`prove` sin(`a`+`b`)−sin(`a`−`b`)=2sin(`a`)sin(`b`)
prouver sec(x)+1=(tan^2(x))/(sec(x)-1)	`prove` sec(`x`)+1=tan²(`x`)sec(`x`)−1
prouver (sin(x))/(1-cos^2(x))=cos(x)	`prove` sin(`x`)1−cos²(`x`) =cos(`x`)
prouver sin^2(x)-cos^2(x)=2(sin^2(x))-1	`prove` sin²(`x`)−cos²(`x`)=2(sin²(`x`))−1
prouver sin^2(3x)=9sin^3(x)cos^3(x)	`prove` sin²(3`x`)=9sin³(`x`)cos³(`x`)
prouver sin^2(x)+cos(-2x)=cos^2(x)	`prove` sin²(`x`)+cos(−2`x`)=cos²(`x`)
prouver sin(pi/2+a)=cos(a)	`prove` sin(π2 +`a`)=cos(`a`)
prouver 2sin^2(x)-cos(x)-2=0	`prove` 2sin²(`x`)−cos(`x`)−2=0
prouver csc(t)-sin(t)=cot(t)*cos(t)	`prove` csc(`t`)−sin(`t`)=cot(`t`)· cos(`t`)
prouver (tan(θ)+6)/(sec(θ))=6cos(θ)+sin(θ)	`prove` tan(θ)+6sec(θ) =6cos(θ)+sin(θ)

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