Question

Cot30/sec30 cosec30/tan45-2cos/sin30 cos^2 45 without using value

Answer 1

$\left(\frac{cot\:30\degree}{sec\:30\degree}\right)\left(\frac{cosec\:30\degree}{tan\:45\degree}\right) - 2\left(\frac{cosec\:30\degree}{sin\:30\degree}\right)\times cos^{2} \:45\degree$

$=\left(\frac{\frac{cos\:30\degree}{sin\:30\degree}}{\frac{1}{cos\:30\degree}}\right)\left(\frac{\frac{1}{sin\:30\degree}}{tan\:45\degree}\right) - 2\left(\frac{\frac{1}{sin\:30\degree}}{sin\:30\degree}\right)\times cos^{2} \:45\degree$

$= \frac{cos^{2}\:30\degree}{sin^{2}\:30\degree\times tan\:45\degree} - 2\times \frac{1}{sin^{2}\:30\degree} \times cos^{2} \:45\degree$

$= \frac{cot^{2} \:30\degree }{tan\:45\degree} - 2\times \frac{1}{sin^{2}\:30\degree} \times cos^{2} \:45\degree$

$= \frac{(\sqrt{3})^{2}}{1} - 2 \times \frac{1}{\left(\frac{1}{2}\right)^{2}} \times \left(\frac{1}{\sqrt{2}}\right)^{2}$

$= \frac{3}{1} - 2 \times 2^{2} \times \frac{1}{2}$

$= 3 - 4 \\= -1$

Therefore.,

$\red {\left(\frac{cot\:30\degree}{sec\:30\degree}\right)\left(\frac{cosec\:30\degree}{tan\:45\degree}\right) - 2\left(\frac{cosec\:30\degree}{sin\:30\degree}\right)\times cos^{2} \:45\degree}\green {= -1}$

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