Question

Evaluate: $\mathtt{\dfrac{2\:tan^2\:45^{\circ}+3\:cos^2\:30^{\circ}-sin^2\:60^{\circ}}{sin^2\:45^{\circ}+cos^2\: 45^{\circ}}}$

Answer 1

Question:-

• Evaluate the given expression

Answer:-

• The value obtained after evaluating the expression is 7/2

Explanation:-

Given :-

$\bullet \; {\underline{\boxed{ \tt { \frac{2 \tan^2 45^\circ + 3 \cos^2 30^\circ - \sin^2 60^\circ}{\sin^2 45^\circ + \cos^2 45^\circ} }}}}$

To Find :-

• The simplified form

Knowledge Required:-

• The value of tan 45° is 1
• The value of cos 30° is √3/2
• The value of sin 60° is √3/2
• The value of sin 45° is 1/√2
• The value of cos 45° is 1/√2

Required Answer :-

• Substituting the above value in the expression

$:{\implies } \tt \dfrac{2 tan^2 45^\circ + 3 cos^2 30^\circ - sin^2 60^\circ}{sin^2 45^\circ + cos^2 45^\circ}$

$:{\implies } \tt \dfrac{2 (1)^2 + 3 \bigg\{ \dfrac{\sqrt 3}{2} \bigg\} ^2 - \bigg\{ \dfrac{\sqrt 3}{2} \bigg\} ^2 }{\bigg\{ \dfrac{ 1}{\sqrt2} \bigg\} ^2 +\bigg\{ \dfrac{ 1}{\sqrt2} \bigg\} ^2 }$

$:{\implies } \tt \dfrac{ 2 + 3 \bigg\{ \dfrac{ 3}{4} \bigg\} - \bigg\{ \dfrac{ 3}{4} \bigg\} }{\bigg\{ \dfrac{ 1}{2} \bigg\} +\bigg\{ \dfrac{ 1}{2} \bigg\} }$

$:{\implies } \tt \dfrac{ \dfrac{8}{4} + \bigg\{ \dfrac{ 9}{4} \bigg\} - \bigg\{ \dfrac{ 3}{4} \bigg\} }{\bigg\{ \dfrac{ 1}{2} \bigg\} +\bigg\{ \dfrac{ 1}{2} \bigg\} }$

$:{\implies } \tt \dfrac{ \bigg\{ \dfrac{8 + 9 - 3}{4} \bigg\} }{\bigg\{ \dfrac{ 1 + 1}{2} \bigg\} }$

$:{\implies } \tt \dfrac{ \bigg\{ \dfrac{14}{4} \bigg\} }{\bigg\{ \dfrac{ 2}{2} \bigg\} }$

$:{\implies } \tt \dfrac{ 14 }{ 4}$

$:{\implies } {\pink{\underline{\boxed{\tt{\dfrac{ 7 }{ 2} }}}}\bigstar}$

Henceforth :-

• The value of the expression is 7/2

Trigonometry Table :-

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

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬