Question

Prove that cos² 60° + sin ²30° + tan²60° + tan²30º =3 5/6​

Answer 1

Given to prove that:-

$\bullet \: \rm cos {}^{2} 60 {}^{ \circ} + sin {}^{2} 30 {}^{ \circ} + tan {}^{2} 60 {}^{ \circ} + tan {}^{2} 30 {}^{ \circ} = 3 \dfrac{5}{6}$

Solution:-

To solve these type of Questions You should know about trigonometric values

sin30° = 1/2

cos60° = 1/2

tan60° = √3

tan30° = 1/√3

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¤Substituting the values

$\dashrightarrow \rm \: \bigg( \dfrac{1}{2} \bigg) {}^{2} + \bigg( \dfrac{1}{2} \bigg) {}^{2} + \bigg( \sqrt{3} \bigg) {}^{2} + \bigg( \dfrac{1}{ \sqrt{3} } \bigg) {}^{2}$

$\dashrightarrow \rm \: \bigg( \dfrac{1}{2 {}^{2} } \bigg) + \bigg( \dfrac{1}{2 {}^{2} } \bigg) + \bigg( 3 \bigg) + \bigg( \dfrac{1}{ (\sqrt{3} ){}^{2} } \bigg)$

$\dashrightarrow \rm \: \bigg( \dfrac{1}{4 } \bigg) + \bigg( \dfrac{1}{4} \bigg) + \bigg( 3 \bigg) + \bigg( \dfrac{1}{ 3 } \bigg)$

$\rm \bigg( \dfrac{1}{4} + \dfrac{1}{4} + 3 + \dfrac{1}{3} \bigg)$

Taking L.C.M to the denominators .

L.C.M of 4,3 is 12

$\dashrightarrow \rm \bigg( \dfrac{3 + 3 + 36 + 4}{12} \bigg)$

$\dashrightarrow \rm \bigg( \cancel\dfrac{46}{12} \bigg)$

$\dashrightarrow \: \bigg( \dfrac{23}{6} \bigg) = 3 \dfrac{5}{6}$

${\boxed{\underline{ \bullet \: \rm cos {}^{2} 60 {}^{ \circ} + sin {}^{2} 30 {}^{ \circ} + tan {}^{2} 60 {}^{ \circ} + tan {}^{2} 30 {}^{ \circ} = 3 \dfrac{5}{6} }}}$

Hence proved !

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