Question:
$\cot ^{2} 30^{\circ}-2 \cos ^{2} 30^{\circ}-\frac{3}{4} \sec ^{2} 45^{\circ}+\frac{1}{4} \operatorname{cosec}^{2} 30^{\circ}$
Solution:
On substituting the values of various T-ratios, we get:
$\cot ^{2} 30^{\circ}-2 \cos ^{2} 30^{\circ}-\frac{3}{4} \sec ^{2} 45^{\circ}+\frac{1}{4} \operatorname{cosec}^{2} 30^{\circ}$
$=(\sqrt{3})^{2}-2 \times\left(\frac{\sqrt{3}}{2}\right)^{2}-\frac{3}{4} \times(\sqrt{2})^{2}+\frac{1}{4} \times(2)^{2}$
$=3-2 \times \frac{3}{4}-\frac{3}{4} \times 2+\frac{1}{4} \times 4$
$=3-\frac{3}{2}-\frac{3}{2}+1$
$=4-\left(\frac{3}{2}+\frac{3}{2}\right)$
$=4-3$
$=1$
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