Question

Solve This:-

Explain in briefly

$\\ \rm \frac{1-sin^2\frac{\pi}{6}}{1+sin^2\frac{\pi}{4}}\times \frac{cos^2\frac{\pi}{3}+cos^2\frac{\pi}{6}}{csc^2\frac{\pi}{2}-cot^2\frac{\pi}{2}} \div \left( sin\frac{\pi}{3}tan\frac{\pi}{6} \right)+\left( sec^2\frac{\pi}{6}-tan^2\frac{\pi}{6} \right)$

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Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm \dfrac{1-sin^2\dfrac{\pi}{6}}{1+sin^2\dfrac{\pi}{4}}\times \dfrac{cos^2\dfrac{\pi}{3}+cos^2\dfrac{\pi}{6}}{csc^2\dfrac{\pi}{2}-cot^2\dfrac{\pi}{2}} \div \left( sin\frac{\pi}{3}tan\frac{\pi}{6} \right)+\left( sec^2\dfrac{\pi}{6}-tan^2\dfrac{\pi}{6} \right)$

Let we solve each term separately first.

Consider

$\rm \: \dfrac{1-sin^2\dfrac{\pi}{6}}{1+sin^2\dfrac{\pi}{4}}$

We know,

$\boxed{\sf\: {sin}^{2}x + {cos}^{2}x = 1 \:}$

So, using this identity, we get

$\rm \: \dfrac{cos^2\dfrac{\pi}{6}}{1+sin^2\dfrac{\pi}{4}}$

On substituting the values of Trigonometric table of standard angles, we have

$\rm \: = \: \dfrac{ {\bigg[\dfrac{ \sqrt{3} }{2} \bigg]}^{2} }{1 + {\bigg[\dfrac{1}{ \sqrt{2} } \bigg]}^{2} }$

$\rm \: = \: \dfrac{\dfrac{3}{4} }{1 + \dfrac{1}{2} }$

$\rm \: = \: \dfrac{\dfrac{3}{4} }{ \dfrac{2 + 1}{2} }$

$\rm \: = \: \dfrac{\dfrac{3}{4} }{ \dfrac{3}{2} }$

$\rm \: = \: \dfrac{3}{4} \times \dfrac{2}{3}$

$\rm \: = \: \dfrac{1}{2}$

Now, Consider

$\rm \: \dfrac{cos^2\dfrac{\pi}{3}+cos^2\dfrac{\pi}{6}}{csc^2\dfrac{\pi}{2}-cot^2\dfrac{\pi}{2}} \div \left( sin\dfrac{\pi}{3}tan\dfrac{\pi}{6} \right)$

We know,

$\boxed{\tt{ {csc}^{2}x - {cot}^{2}x \: = \: 1 \: }}$

and

$\boxed{\tt{ sin\bigg(\dfrac{\pi}{2} - x\bigg) = cosx \: }}$

So, using these Identities, we get

can be rewritten as

$\rm \: = \: \dfrac{cos^2\dfrac{\pi}{3}+sin^2\bigg(\dfrac{\pi}{2} - \dfrac{\pi}{6}\bigg)}{1} \div \bigg(\dfrac{ \sqrt{3} }{2} \times \dfrac{1}{ \sqrt{3} } \bigg)$

$\rm \: = \: \dfrac{cos^2\dfrac{\pi}{3}+sin^2\bigg(\dfrac{\pi}{3} \bigg)}{1} \div \bigg(\dfrac{1}{2}\bigg)$

We know,

$\boxed{\sf\: {sin}^{2}x + {cos}^{2}x = 1 \:}$

So, using this, we get

$\rm \: = \: 1 \div \dfrac{1}{2}$

$\rm \: = \: 1 \times 2$

$\rm \: = \: 2$

Now, Consider

$\rm \: sec^2\dfrac{\pi}{6}-tan^2\dfrac{\pi}{6}$

We know,

$\boxed{\tt{ \: {sec}^{2}x - {tan}^{2}x \: = \: 1 \: }}$

So, using this, we get

$\rm \: = \: 1$

Now, Consider the given expression

$\rm \dfrac{1-sin^2\dfrac{\pi}{6}}{1+sin^2\dfrac{\pi}{4}}\times \dfrac{cos^2\dfrac{\pi}{3}+cos^2\dfrac{\pi}{6}}{csc^2\dfrac{\pi}{2}-cot^2\dfrac{\pi}{2}} \div \left( sin\dfrac{\pi}{3}tan\dfrac{\pi}{6} \right)+\left( sec^2\dfrac{\pi}{6}-tan^2\dfrac{\pi}{6} \right)$

$\rm \: = \: \dfrac{1}{2} \times 2 + 1$

$\rm \: = \: 1 + 1$

$\rm \: = \: 2$

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$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf0 & \bf\dfrac{\pi}{6} & \bf\dfrac{\pi}{4} & \bf\dfrac{\pi}{6} & \bf\dfrac{\pi}{2} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

Answer 2

Answer

$\rm \frac{1-sin^2\frac{\pi}{6}}{1+sin^2\frac{\pi}{4}}\times \frac{cos^2\frac{\pi}{3}+cos^2\frac{\pi}{6}}{csc^2\frac{\pi}{2}-cot^2\frac{\pi}{2}} \div \left( sin\frac{\pi}{3}tan\frac{\pi}{6} \right)+\left( sec^2\frac{\pi}{6}-tan^2\frac{\pi}{6} \right)\\\\:\longrightarrow \rm \frac{1-sin^2(30)}{1+sin^2(45)}\times \frac{cos^2(60)+cos^2(30)}{csc^2(90)-cot^2(90)}\div (sin(60).tan(30)+(sec^2(30)-tan^2(30))$

$\large\sf:\to \rm \frac{1-\frac{1}{4}}{1+\frac{1}{2}}\times \frac{\frac{1}{4}+\frac{3}{4}}{1-0}\div \bigg(\frac{\sqrt{3}}{2}.\frac{1}{\sqrt{3}} \bigg)+ \bigg(\frac{4}{3}-\frac{1}{3} \bigg)\\\\\large \sf:\longrightarrow \: \rm \frac{\frac{3}{4}}{\frac{3}{2}}\times \frac{1}{1}\div \bigg( \: \: \frac{1}{2} \: \: \bigg)+(1)\\\\:\large \longrightarrow\rm \frac{1}{2}\times (2)+1\\\\:\to\large \rm 1+1\\\\:\to\large \rm 2$

Danger points

• → Be aware of Bodmas rule

where:-

• B denotes Brackets

• O denotes of

• D denotes division

• M denotes multiplication

• A denotes addition

• S denotes subtraction