Question

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sin²30°cos²45°+4tan²30°+½sin²90°+1/8cot²60°

Evaluate


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Answer 1

I'm not a moderator or a star but hope it helps you DEAR! :)

*see the picture for better explanations*

Answer 2

Answer:

The value of the given expression is 2

Step-by-step explanation:

$\sf{sin^{2}30^{\circ}cos^{2}45^{\circ}+4tan^{2}30^{\circ}+\dfrac{1}{2}sin^{2}90^{\circ}+\dfrac{1}{8}cot^{2}60^{\circ}}$

$\implies \sf{\bigg(\dfrac{1}{2}\bigg)^{2}\times \bigg(\dfrac{1}{\sqrt{2}}\bigg)^{2}+4\bigg(\dfrac{1}{\sqrt{3}}\bigg)^{2}+\dfrac{1}{2}(1)^{2}+\dfrac{1}{8}\bigg(\dfrac{1}{\sqrt{3}}\bigg)^{2}}$

$\implies \sf{\dfrac{1}{4}\times\dfrac{1}{2}+4\bigg(\dfrac{1}{3}\bigg)+\dfrac{1}{2}+\dfrac{1}{8}\times\dfrac{1}{3}}$

$\implies \sf{\dfrac{1}{8}+\dfrac{4}{3}+\dfrac{1}{2}+\dfrac{1}{24}}$

On taking LCM,

$\implies \sf{\dfrac{3+(4\times 8)+12+1}{24}}$

$\implies \sf{\dfrac{3+32+12+1}{24}}$

$\implies \sf{\dfrac{48}{24}}$

$\implies \sf{2}$

$\therefore \: \boxed{\bf{sin^{2}30^{\circ}cos^{2}45^{\circ}+4tan^{2}30^{\circ}+\dfrac{1}{2}sin^{2}90^{\circ}+\dfrac{1}{8}cot^{2}60^{\circ}}=2}$

Below is the trigonometry value table which has to be learnt and can be applied for questions similar to this.

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