Step-by-step explanation:
Consider,
[tex]\qquad\sf \: {tan}^{2}A - {sin}^{2}A \\ \\ [/tex]
can be rewritten as
[tex]\qquad\sf \: = \: \dfrac{ {sin}^{2} A}{ {cos}^{2} A} - {sin}^{2}A \\ \\ [/tex]
[tex]\qquad\sf \: = \:\left( \dfrac{1}{ {cos}^{2} A} - 1\right) {sin}^{2}A \\ \\ [/tex]
[tex]\qquad\sf \: = \:\left( {sec}^{2}A - 1\right) {sin}^{2}A \\ \\ [/tex]
[tex]\qquad\sf \: = \: ({tan}^{2}A ) {sin}^{2}A \\ \\ [/tex]
[tex]\qquad\sf \: = \: {tan}^{2}A \: {sin}^{2}A \\ \\ [/tex]
Hence,
[tex]\qquad\sf\implies \bf \: {tan}^{2}A - {sin}^{2}A = {tan}^{2}A \: {sin}^{2}A \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Formulae Used
[tex]\sf \:tanx \: = \: \dfrac{sinx}{cosx} \\ \\ [/tex]
[tex]\sf \:secx \: = \: \dfrac{1}{cosx} \\ \\ [/tex]
[tex]\sf \: {sec}^{2}x - {tan}^{2}x = 1 \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: Formulae}}}} \\ \\ \bigstar \: \bf{sinx = \dfrac{1}{cosecx} }\\ \\ \bigstar \: \bf{cosx = \dfrac{1}{secx} }\\ \\ \bigstar \: \bf{tanx = \dfrac{sinx}{cosx} = \dfrac{1}{cotx} }\\ \\ \bigstar \: \bf{cot x= \dfrac{cosx}{sinx} = \dfrac{1}{tanx} }\\ \\ \bigstar \: \bf{cosec x) = \dfrac{1}{sinx} }\\ \\ \bigstar \: \bf{secx = \dfrac{1}{cosx} }\\ \\ \bigstar \: \bf{ {sin}^{2}x + {cos}^{2}x = 1 } \\ \\ \bigstar \: \bf{ {sec}^{2}x - {tan}^{2}x = 1 }\\ \\ \bigstar \: \bf{ {cosec}^{2}x - {cot}^{2}x = 1 } \: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]
