Use the special angles to evaluate: a) \( \operatorname{cosec} 60^{\circ} \cdot \cot 30^{\circ}+\cos 45^{\circ} \cdot \operatorname{cosec} 45^{\circ} \) b) \( \sec ^{2} 45^{\circ}-\tan ^{2} 45^{\circ} \) c) \( 1+\cot ^{2} 60^{\circ} \) d) \( \operatorname{cosec}^{2} 60^{\circ}+\cot ^{2} 45^{\circ}+\sec ^{2} 30^{\circ} \) e) \( \sin 90^{\circ}-\cos 60^{\circ} \cdot \sec 60^{\circ}+\operatorname{cosec} 45^{\circ} \cdot \cos 45^{\circ} \) f) \( \operatorname{cosec} 60^{\circ} \cdot \tan 45^{\circ} \cdot \sin 30^{\circ} \)
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To evaluate the given expressions, we can utilize the known values of the trigonometric ratios for special angles: a) \( \operatorname{cosec} 60^{\circ} \cdot \cot 30^{\circ} + \cos 45^{\circ} \cdot \operatorname{cosec} 45^{\circ} = \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{3} + \frac{1}{\sqrt{2}} \cdot \sqrt{2} = \frac{2}{3} + 1 = \frac{5}{3} \). b) \( \sec^{2} 45^{\circ} - \tan^{2} 45^{\circ} = 2 - 1 = 1 \). c) \( 1 + \cot^{2} 60^{\circ} = 1 + \frac{1}{\sqrt{3}}^{2} = 1 + \frac{1}{3} = \frac{4}{3} \). d) \( \operatorname{cosec}^{2} 60^{\circ} + \cot^{2} 45^{\circ} + \sec^{2} 30^{\circ} = \frac{4}{3} + 1 + 4 = \frac{4}{3} + \frac{3}{3} + \frac{12}{3} = \frac{19}{3} \). e) \( \sin 90^{\circ} - \cos 60^{\circ} \cdot \sec 60^{\circ} + \operatorname{cosec} 45^{\circ} \cdot \cos 45^{\circ} = 1 - \frac{1}{2} \cdot \frac{2}{\sqrt{3}} + 1 \cdot \frac{1}{\sqrt{2}} = 1 - \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{2}} \). f) \( \operatorname{cosec} 60^{\circ} \cdot \tan 45^{\circ} \cdot \sin 30^{\circ} = \frac{2}{\sqrt{3}} \cdot 1 \cdot \frac{1}{2} = \frac{1}{\sqrt{3}} \). So the evaluations yield the following results: a) \( \frac{5}{3} \) b) \( 1 \) c) \( \frac{4}{3} \) d) \( \frac{19}{3} \) e) \( 1 - \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{2}} \) f) \( \frac{1}{\sqrt{3}} \)
