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  • Prove the following : sin theta by 1+ cos theta + 1+cos theta by sin theta = 2 cosec theta

Prove the following

:\frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta}{\sin \theta}= 2\cos ec\theta

Answers (1)

\frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta}{\sin \theta}= 2\cos ec\theta

Taking L.H.S.

 = \frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta}{\sin \theta}

Taking LCM
 = \frac{\sin ^{2}\theta +\left ( 1+\cos \theta \right )^{2}}{\left ( 1+\cos \theta \right )\sin \theta }
 = \frac{\sin ^{2}\theta +\cos ^{2}\theta +2\cos \theta }{\left ( 1+\cos \theta \right )\sin \theta }          \left ( \because \left ( a+b \right )^{2} = a^{2}+b^{2}+2ab\right )
 = \frac{1+1+2\cos \theta }{\left ( 1+\cos \theta \right )\sin \theta }                            \left ( \because \sin ^{2}\theta +\cos ^{2}\theta = 1 \right )
 = \frac{2\left ( 1+\cos \theta \right )}{\left ( 1+\cos \theta \right )\sin \theta }
 = \frac{2}{\sin \theta }
= 2 \cos ec\theta         \left ( \because \frac{1}{\sin \theta } = \cos ec \theta \right )
         

L.H.S. = R.H.S.
Hence proved.      
               

Posted by

infoexpert27

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