Saturday, 13 July 2019

trigonometry - Proving the identity $csc x−sin x = (cot x)(cos x) $



We recently started Trigonometry and I was trying to solve this.



$$\csc x − \sin x=(\cot x)(\cos x) $$



So starting with LHS:
\begin{align}
\frac{1}{\sin x} − \sin x &= \frac{1 − (\sin x)^2}{ \sin x } \\

&= \frac{(\cos x)^2 }{ \sin x }
\end{align}



I am stuck now and wanted to know how should I proceed. Is this much correct?


Answer



Its done. Just split the numerator as:



$$ \frac{\cos x . \cos x }{\sin x}$$



$$= \frac{\cos x }{\sin x} . \cos x $$




$$= \cot x . \cos x $$



Proved!


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