contestada

Prove the identity.

cotx(1-cos2x)=sin2x

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Respuesta :

WonderlandWanderer

Prove that

[tex]cot(x)*(1-cos(2x))=sin(2x)[/tex]

Proof:

  • [tex]cot(x)=\frac{cos(x)}{sin(x)}[/tex]

⇒ [tex]\frac{cos(x)}{sin(x)} *(1-cos(2x))=sin(2x)[/tex]

  • [tex]sin(2x)=2sin(x)cos(x)[/tex]
  • [tex]cos(2x)=cos^2(x)-sin^2(x)[/tex]

⇒ [tex]\frac{cos(x)}{sin(x)}*(1-(cos^2(x)-sin^2(x)))=2*sin(x)*cos(x)[/tex]

⇒ [tex]\frac{cos(x)}{sin(x)}*(1-cos^2(x)+sin^2(x))=2*sin(x)*cos(x)[/tex]

  • Multiply both sides by [tex]sin(x)[/tex], [tex]sin(x)[/tex] ≠ 0

⇒ [tex]cos(x)*(1-cos^2(x)+sin^2(x))=2*sin(x)*cos(x)*sin(x)[/tex]

⇒ [tex]cos(x)*(1-cos^2(x)+sin^2(x))=2*sin^2(x)*cos(x)[/tex]

⇒ [tex]cos(x) - cos^3(x)+cos(x)*sin^2(x)=2*sin^2(x)*cos(x)[/tex]

⇒ [tex]cos(x)*(1-cos^2(x)) + cos(x)*sin^2(x)=2*sin^2(x)*cos(x)[/tex]

  • [tex]1-cos^2(x) = sin^2(x)[/tex]

⇒ [tex]cos(x)*sin^2(x) + cos(x)*sin^2(x)=2*sin^2(x)*cos(x)[/tex]

⇒ [tex]2*cos(x)*sin^2(x) = 2*sin^2(x)*cos(x)[/tex]

⇒ [tex]2*cos(x)*sin^2(x) = 2*cos(x)*sin^2(x)[/tex]

Q.E.D.

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