Question

$\tiny\tt\colorbox{aqua}{ꌚꂦ꒒꒦ꏂ ꓄ꀍꀤꌚ ꁅꀎꌩꌚ}$
$\tiny{(1 + \cot(o) + \tan(o) )( \sin(o) - \cos(o) ) = \frac{ \sec(o) }{cosec {}^{2}o } - \frac{cosec (o)}{ \sec {}^{2} (o) } }$
$\tiny \bold{@DarkShadow7083 }$
Everything is tiny! Tiny girl! Prove the identity given above-​

Answer 1

$\text{\bf To prove, }\\\text{\: \: \: (1 + \cot \theta + \tan \theta)( \sin \theta - \cos \theta) = \dfrac{ \sec \theta}{\csc^2 \theta} - \dfrac{\csc \theta}{\sec^2 \theta} }$

$\text{\bf Solution}$

  $\textsf{Taking,}\\\texttt{\: \: LHS = }(1 + \cot \theta + \tan \theta) (\sin \theta - \cos \theta)\\\\\texttt{\hspace{3em} = \sin \theta - \cos \theta + \cot \theta \sin \theta - \cot \theta \cos \theta + \tan \theta \sin \theta - \tan \theta \cos \theta }\\\\\texttt{\hspace{3em} = \sin \theta - \cos \theta + \dfrac{\cos \theta}{\sin \theta} \cdot \sin \theta - \cot \theta \cos \theta + \tan \theta \sin \theta - \dfrac{\sin \theta}{\cos \theta} \cdot \cos \theta }$

               $\texttt{ = \sin \theta - \cos \theta + \cos \theta - \cot \theta \cos \theta + \tan \theta \sin \theta - \sin \theta }\\\\\texttt{ = \sin \theta \tan \theta - \cot \theta \cos \theta }\\\\\texttt{ = \dfrac{\sin \theta}{\cos \theta} \cdot \dfrac{1}{\csc \theta} + \dfrac{\cos \theta}{\sin \theta} \cdot \dfrac{1}{\sec \theta} }\\\\\texttt{ = \dfrac{1}{\csc \theta} \cdot \dfrac{1}{\csc \theta} \cdot \sec \theta + \dfrac{1}{\sec \theta} \cdot \dfrac{1}{\sec \theta} \cdot \csc \theta }$

               $\texttt{ = \dfrac{\sec \theta}{\csc^2 \theta} - \dfrac{\csc \theta}{\sec^2 \theta} }\\\\\texttt{ = R.H.S}$

       $\textsf{Hence, proved.}$

Answer 2

$\huge \underline \mathfrak{⇛to \: prove} \\ \\ {(1 + \cot(o) + \tan(o) )( \sin(o) - \cos(o) ) = \frac{ \sec(o) }{cosec {}^{2}o } - \frac{cosec (o)}{ \sec {}^{2} (o) } } \\ \\ \\ ⇛\huge \underline \mathfrak{proof} \\ \\ LHS: (1 + \frac{ \cos(o) }{ \sin(o) } + \frac{ \cos(o) }{ \sin(o) })( \sin(o) - \cos(o) ) \\ \\ ⇛\frac{( \sin(o) \cos(o) + \cos {}^{2} (o) + \sin {}^{2} (o) ) ( \sin(o) - \cos(o) ) }{ \sin(o) \cos(o) } \\ \\ ⇛\frac{ \sin {}^{3} (o) - \cos {}^{3} (o) }{ \sin(o) \cos(o) } \\ \\ ⇛ \frac{ \sin {}^{2} (o) }{ \cos(o) } - \frac{ \cos {}^{2} (o) }{ \sin(o) } \\ \\ ⇛ \frac{ \sec(o) }{ \csc {}^{2} (o) } - \frac{ \csc(o) }{ \sec {}^{2} (o) } \\ \\ \: \: \: \: \: \: \: \huge \bold{@WildCat7083 }$