Question

Show That 4 (sin⁴ 30⁰+ cos⁴ 60⁰) -3 (cos² 45⁰- sin² 90⁰)=2​

Answer 1

$\huge{\underline{\underline{\bf{\blue{Given\::}}}}}$

• $\sf{4\:(sin^{4}\:30°\:+\:cos}^{4}\:{60°)\:-\:3\:(cos}^{2}\:{45°\:-\:sin}^{2}\:{90°\:)\:=\:2}$

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$\huge{\underline{\underline{\bf{\blue{Show\:that\::}}}}}$

• $\sf{4\:(sin^{4}\:30°\:+\:cos}^{4}\:{60°)\:-\:3\:(cos}^{2}\:{45°\:-\:sin}^{2}\:{90°\:)\:=\:2}$

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$\huge{\underline{\underline{\bf{\blue{Solution\::}}}}}$

$\:\:\:\:\:\::\:\Longrightarrow\footnotesize\sf\:{4\:(sin^{4}\:30°\:+\:cos}^{4}\:{60°)\:-\:3\:(cos}^{2}\:{45°\:-\:sin}^{2}\:{90°\:)\:=\:{\purple{2}}}$

$\:\:\:\:\:\::\:\Longrightarrow\footnotesize\sf\:{4\:{\Large{[}}\:(\:{\frac{1}{2}}\:)^{4}\:+\:(\:{\frac{1}{2}}\:)^{4}\:{\Large{]}}\:-\:3\:{\Large{[}}\:(\:{\frac{1}{\sqrt{2}}}\:)^{2}\:-\:(\:1\:)}^{2}\:{\Large{]}}\:=\:{\purple{2}}$

$\:\:\:\:\:\::\:\Longrightarrow\sf\:{4\:{\Large{[}}\:{\frac{1}{16}}\:+\:{\frac{1}{16}}{\Large{]}}\:-\:3\:{\Large{[}}\:{\frac{1}{2}}\:-\:1\:{\Large{]}}\:=\:{\purple{2}}}$

$\:\:\:\:\:\::\:\Longrightarrow\sf\:{4\:{\Large{[}}\:{\frac{1\:+\:1}{16}}{\Large{]}}\:-\:3\:{\Large{[}}\:{\frac{1\:-\:2}{2}}{\Large{]}}\:=\:{\purple{2}}}$

$\:\:\:\:\:\::\:\Longrightarrow\sf\:{4\:\times\:{\frac{2}{16}}\:-\:3\:\times\:{\frac{-1}{2}}\:=\:{\purple{2}}}$

$\:\:\:\:\:\::\:\Longrightarrow\sf\:{\frac{1}{2}}\:+\:{\frac{3}{2}}\:=\:{\purple{2}}$

$\:\:\:\:\:\::\:\Longrightarrow\sf\:{\frac{3\:+\:1}{2}}\:=\:{\purple{2}}$

$\:\:\:\:\:\::\:\Longrightarrow\sf\:{\frac{4}{2}}\:=\:{\purple{2}}$

$\:\:\:\:\:\::\:\Longrightarrow\sf\:{2\:=\:{\purple{2}}}$

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$\:\:\:\:\:\:{\bold{\pink{\dag}}}\:{\sf{\underline{So,\:it\:is\:showed\:{\green{\mathfrak{\bold{2\:=\:2}}}}}}}$

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Know More:

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$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0 \end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}$