(a) Evaluate: (5 cos^2 60^°+4 sec^2 30^°- tan^2 45^°)/(sin^2 30^°+ cos^2 30^°)

\(\displaystyle \frac{5 \cos ^{2} 60^{\circ}+4 \sec ^{2} 30^{\circ}-\tan ^{2} 45^{\circ}}{\sin ^{2} 30^{\circ}+\cos ^{2} 30^{\circ}}\)

\(\displaystyle =\frac{5\left(\dfrac{1}{2}\right)^{2}+4\left(\dfrac{2}{\sqrt{3}}\right)^{2}-(1)^{2}}{1}\)

\(\displaystyle =\frac{67}{12}\)

LHS \(\displaystyle =1+\frac{\cot ^{2} \alpha}{1+\operatorname{cosec} \alpha}\)

\(\displaystyle =1+\frac{(\operatorname{cosec} \alpha+1)(\operatorname{cosec} \alpha-1)}{(1+\operatorname{cosec} \alpha)}\)

\(\displaystyle =1+\operatorname{cosec} \alpha-1\)

\(\displaystyle =\operatorname{cosec} \alpha=\) RHS