Question

Prove the identity.

cotx(1-cos2x)=sin2x

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Answer 1

Prove that

`cot(x)*(1-cos(2x))=sin(2x)`

Proof:

• 
  `cot(x)=(cos(x))/(sin(x))`

⇒
`(cos(x))/(sin(x)) *(1-cos(2x))=sin(2x)`

• 
  `sin(2x)=2sin(x)cos(x)`
• 
  `cos(2x)=cos^2(x)-sin^2(x)`

⇒
`(cos(x))/(sin(x))*(1-(cos^2(x)-sin^2(x)))=2*sin(x)*cos(x)`

⇒
`(cos(x))/(sin(x))*(1-cos^2(x)+sin^2(x))=2*sin(x)*cos(x)`

• Multiply both sides by
  `sin(x)`,
  `sin(x)` ≠ 0

⇒
`cos(x)*(1-cos^2(x)+sin^2(x))=2*sin(x)*cos(x)*sin(x)`

⇒
`cos(x)*(1-cos^2(x)+sin^2(x))=2*sin^2(x)*cos(x)`

⇒
`cos(x) - cos^3(x)+cos(x)*sin^2(x)=2*sin^2(x)*cos(x)`

⇒
`cos(x)*(1-cos^2(x)) + cos(x)*sin^2(x)=2*sin^2(x)*cos(x)`

• 
  `1-cos^2(x) = sin^2(x)`

⇒
`cos(x)*sin^2(x) + cos(x)*sin^2(x)=2*sin^2(x)*cos(x)`

⇒
`2*cos(x)*sin^2(x) = 2*sin^2(x)*cos(x)`

⇒
`2*cos(x)*sin^2(x) = 2*cos(x)*sin^2(x)`

Q.E.D.