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Question

Solve:$$ \dfrac{-\sin \theta}{1-\cos \theta}=-\cot \dfrac{\theta }{2}$$ 


Solution

LHS $$=\dfrac { -\sin\theta  }{ 1-\cos\theta  } \\ =\dfrac { -\sin2\left( \dfrac { \theta  }{ 2 }  \right)  }{ 1-\cos2\left( \dfrac { \theta  }{ 2 }  \right)  } \\ =-\dfrac { 2\sin\left( \dfrac { \theta  }{ 2 }  \right) \cos\left( \dfrac { \theta  }{ 2 }  \right)  }{ 2{ \sin }^{ 2 }\dfrac { \theta  }{ 2 }  } \\ =-\dfrac { \cos\left( \dfrac { \theta  }{ 2 }  \right)  }{ \sin\left( \dfrac { \theta  }{ 2 }  \right)  } \\ =-\cot\left( \dfrac { \theta  }{ 2 }  \right) $$
        $$=$$ RHS


Mathematics

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