Courses
Courses for Kids
Free study material
Free LIVE classes
More
Questions & Answers
seo-qna

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

Answer
VerifiedVerified
330.3k+ views
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.

Last updated date: 03rd Jun 2023
•
Total views: 330.3k
•
Views today: 7.86k