Question

$\large{\underline{\underline{\sf{\red{Question:}}}}}$
$\sf{Simplify: - }$
$\\ \sf{ \dfrac{1 - {sin}^{2} \dfrac{\pi}{6} }{1 + {sin}^{2} \dfrac{\pi}{4} } \times \dfrac{ {cos}^{2} \dfrac{\pi}{3} + {cos}^{2} \dfrac{\pi}{6} }{ {cosec}^{2} \dfrac{\pi}{2} -{cot}^{2} \dfrac{\pi}{2}} \: \div (sin \frac{\pi}{3} tan \frac{\pi}{6}) + ( {sec}^{2} \dfrac{\pi}{6} - {tan}^{2} \dfrac{\pi}{6} )}$

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Answer 1

$\red{\large\mathfrak {Question}}$

$\sf{ \dfrac{1 - {sin}^{2} \dfrac{\pi}{6} }{1 + {sin}^{2} \dfrac{\pi}{4} } \times \dfrac{ {cos}^{2} \dfrac{\pi}{3} + {cos}^{2} \dfrac{\pi}{6} }{ {cosec}^{2} \dfrac{\pi}{2} -{cot}^{2} \dfrac{\pi}{2}} \: \div (sin \frac{\pi}{3} tan \frac{\pi}{6}) + ( {sec}^{2} \dfrac{\pi}{6} - {tan}^{2} \dfrac{\pi}{6} )}$

$\blue{\large\mathfrak {Solution}}$

A/q to trigonometric ratios : 1-sin²=cos²

Now putting the values of the variables :–

$=\sf \frac{ {cos}^{2} \frac{\pi}{6} }{1 + { (\frac{1}{2} )}^{2} } \times \frac{ ({ \frac{1}{2}) }^{2} + ( { \frac{ \sqrt{3} }{2}) }^{2} }{1} \div ( \frac{ \sqrt{3} }{2} \times \frac{1}{ \sqrt{3} } ) \times 1$

$= \frac{ { (\frac{ \sqrt{3} }{2}) }^{2} }{1 + \frac{1}{2} } \times \frac{ \frac{1}{4} + \frac{3}{4} }{1} \div \frac{1}{2} \times 1$

$= \frac{ \frac{3}{4} }{ \frac{3}{2} } \times \frac{1}{1} \times 2 \times 1$

$= \frac{3}{4} \times \frac{2}{3} \times 2$

Cancelling all the terms, we get :—

$= \bold{\boxed{\pink{\large{ \: \: 1 \: \: }}}}</p><p>$

$\green{\large\mathfrak {Hope \: it \: helps \: you}}$

Answer 2

$Solution</p><p></p><p>A/q to trigonometric ratios : 1-sin²=cos²</p><p></p><p>Now putting the values of the variables :–</p><p></p><p>\begin{gathered} =\sf \frac{ {cos}^{2} \frac{\pi}{6} }{1 + { (\frac{1}{2} )}^{2} } \times \frac{ ({ \frac{1}{2}) }^{2} + ( { \frac{ \sqrt{3} }{2}) }^{2} }{1} \div ( \frac{ \sqrt{3} }{2} \times \frac{1}{ \sqrt{3} } ) \times 1 \\ \end{gathered}=1+(21)2cos26π×1(21)2+(23)2÷(23×31)×1</p><p></p><p>\begin{gathered} = \frac{ { (\frac{ \sqrt{3} }{2}) }^{2} }{1 + \frac{1}{2} } \times \frac{ \frac{1}{4} + \frac{3}{4} }{1} \div \frac{1}{2} \times 1 \\ \end{gathered}=1+21(23)2×141+43÷21×1</p><p></p><p>\begin{gathered} = \frac{ \frac{3}{4} }{ \frac{3}{2} } \times \frac{1}{1} \times 2 \times 1 \\ \end{gathered}=2343×11×2×1</p><p></p><p>\begin{gathered} = \frac{3}{4} \times \frac{2}{3} \times 2 \\ \end{gathered}=43×32×2</p><p></p><p>Cancelling all the terms, we get :—</p><p></p><p>= \bold{\boxed{\pink{\large{ \: \: 1 \: \: }}}} < /p > < p >=1</p><p></p><p></p><p>\green{\large\mathfrak {Hope \: it \: helps \: you}}Hopeithelpsyou</p><p></p><p>$