Question

sin theta/cot theta + cosec theta = 2 + sin theta/cot theta - cosec theta​

Answer 1

$\boxed{ \boxed{\begin{array}{} \sf \rightarrow We \: have \: to \: prove \: that \\\\ \hookrightarrow\rm\dfrac{ \sin \theta}{ \cot \theta + \cosec \theta } = 2 + \dfrac{ \sin \theta}{ \cot \theta - \cosec \theta } or,\dfrac{ \sin \theta}{ \cot \theta + \cosec \theta } - \dfrac{ \sin \theta}{ \cot \theta - \cosec \theta } = 2\end{array}}}$

$\boxed{ \boxed{\begin{array}{ lr} \rm LHS = \dfrac{ \sin \theta}{ \cot \theta + \cosec \theta } - \dfrac{ \sin \theta}{ \cot \theta - \cosec \theta } \\\\\\ \hookrightarrow \dfrac{ \sin \theta}{ \cosec \theta + \cot \theta} + \dfrac{ \sin \theta}{ \cosec \theta - \cot \theta} \\\\\\ \hookrightarrow \sin \theta \bigg \{\dfrac{ 1}{ \cosec \theta + \cot \theta} + \dfrac{ 1}{ \cosec \theta - \cot \theta} \bigg \} \\\\\\ \hookrightarrow \sin \theta \bigg \{\dfrac{ \cosec \theta \cancel{ - \cot \theta} + \cosec \theta \cancel{+ \cot \theta}}{ \cosec ^{2} \theta - \cot ^{2} \theta} \bigg\} \\\\\\ \hookrightarrow \sin \theta \bigg \{ \dfrac{2 \cosec \theta}{1} \bigg \} \\\\\\ \hookrightarrow \sin \theta(2 \cosec \theta) \\\\\\ \hookrightarrow \cancel{ 2\sin \theta} \times \dfrac{1}{ \cancel{\sin \theta}} \\\\\\ \hookrightarrow 2\\ \\ \end{array}}}$

$\boxed{\boxed{ \begin{array}{lr} \boxed{\rm LHS = \dfrac{ \sin \theta}{ \cot \theta + \cosec \theta} } \\\\\\ \hookrightarrow \sin \theta( \cosec \theta - \cot \theta) \\ \\ \\ \hookrightarrow \sin \theta \bigg \lbrace \dfrac{1}{ \sin \theta} - \dfrac{ \cos \theta}{ \sin \theta} \bigg \rbrace \\\\\\\hookrightarrow \cancel{\sin \theta} \bigg \lbrace \dfrac{1 - \cos \theta}{ \cancel{\sin \theta} } \bigg\rbrace \\\\\\ \hookrightarrow 1 - \cos \theta \\\\\\ \hookrightarrow2 - (1 + \cos \theta) \\\\\\ \hookrightarrow2 - \dfrac{(1 + \cos \theta)(1 - \cos \theta)}{1 - \cos \theta} \\\\\\ \hookrightarrow2 - \dfrac{(1 - \cos^{2} \theta)}{1 - \cos \theta} \\\\\\ \hookrightarrow2 - \dfrac{\sin^{2} \theta}{1 - \cos \theta} \\\\\\ \hookrightarrow2 - \dfrac{ \sin \theta}{ \dfrac{1 - \cos \theta}{ \sin \theta} } \\\\\\ \hookrightarrow2 - \dfrac{ \sin \theta}{ \dfrac{1}{ \sin \theta} - \dfrac{ \cos \theta}{ \sin \theta} } \\\\\\ \boxed{ \rm RHS = 2 - \dfrac{ \sin \theta}{ \cosec \theta - \cot \theta} } \\ \\ \end{array}}}$