Prove that: $\frac{{\sin heta }}{{1 - \cos heta }} = cosec heta + \cot heta $

Question

Prove that: $\frac{{\sin 	heta }}{{1 - \cos 	heta }} = cosec	heta  + \cot 	heta $

Vedantu Content Team · Accepted Answer

Hint: - Taking conjugate of denominator.
Given,
L.H.S $\begin{gathered}
   = \frac{{\sin \theta }}{{1 - \cos \theta }} \\
    \\
\end{gathered} $
Multiply and Divide by $\left( {1 + \cos \theta } \right)$ , we get
           $\begin{gathered}
   = \frac{{\sin \theta }}{{1 - \cos \theta }} \times \frac{{1 + \cos \theta }}{{1 + \cos \theta }} \\
   = \frac{{\sin \theta \left( {1 + \cos \theta } \right)}}{{\left( {1 - \cos \theta } \right)\left( {1 + \cos \theta } \right)}} \\
   = \frac{{\sin \theta \left( {1 + \cos \theta } \right)}}{{{{\left( 1 \right)}^2} - {{\left( {\cos \theta } \right)}^2}}} \\
   = \frac{{\sin \theta \left( {1 + \cos \theta } \right)}}{{1 - {{\cos }^2}\theta }} \\
\end{gathered} $
We know that, ${\sin ^2}\theta + {\cos ^2}\theta = 1$
  Or, ${\sin ^2}\theta = 1 - {\cos ^2}\theta $
Replace $\left( {1 - {{\cos }^2}\theta } \right)$ by ${\sin ^2}\theta $ , we get
L.H.S $\begin{gathered}
   = \frac{{\sin \theta \left( {1 + \cos \theta } \right)}}{{{{\sin }^2}\theta }} \\
   = \frac{{\sin \theta \left( {1 + \cos \theta } \right)}}{{\sin \theta \times \sin \theta }} \\
\end{gathered} $
Cancel out $\sin \theta $ in numerator by $\sin \theta $ in denominator, we get
           $\begin{gathered}
   = \frac{{1 + \cos \theta }}{{\sin \theta }} \\
   = \frac{1}{{\sin \theta }} + \frac{{\cos \theta }}{{\sin \theta }} \\
\end{gathered} $
Now we can written $\frac{1}{{\sin \theta }} = \cos ec\theta $ and $\frac{{\cos \theta }}{{\sin \theta }} = \cot \theta $ , we get
   L.H.S $ = \cos ec\theta + \cot \theta $- (1)
Since, given R.H.S$ = \cos ec\theta + \cot \theta $ - (2)
By seeing equation (1) and (2) we can tell that
      L.H.S R.H.S
Hence, it proved.
Note: - These types of questions are also solve by taking R.H.S (Right Hand Side), solve it to prove L.H.S (Left Hand Side). During solving trigonometry proving we should always have basic trigonometry identities in our mind.

Prove that:
$\frac{{\sin \theta }}{{1 - \cos \theta }} = cosec\theta + \cot \theta $

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Prove that: $\frac{{\sin \theta }}{{1 - \cos \theta }} = cosec\theta + \cot \theta $