[tex]a)sin \: 30^{\circ}cos \: 60^{\circ} + sin \: 60^{\circ}cos \: 30^{\circ}[/tex]
[tex] = \frac{1}{2} \times \frac{1}{2} + \frac{ \sqrt{3} }{2} \times \frac{ \sqrt{3} }{2} [/tex]
[tex] = \frac{1 \times 1}{2 \times 2} + \frac{ \sqrt{3} \times \sqrt{3} }{2 \times 2} [/tex]
[tex] = \frac{1}{4} + \frac{3}{4} [/tex]
[tex] = \frac{1 + 3}{4} [/tex]
[tex] = \frac{4}{4} [/tex]
[tex] = 1[/tex]
[tex]b)cos \: 60^{\circ}cos \: 30^{\circ} - sin \: 60^{\circ}sin \: 30^{\circ}[/tex]
[tex] = \frac{1}{2} \times \frac{ \sqrt{3} }{2} - \frac{ \sqrt{3} }{2} \times \frac{1}{2} [/tex]
[tex] = 0[/tex]
[tex]c)cos \: 60^{\circ} - {cos}^{2} 30^{\circ} + {sin}^{2} 30^{\circ}[/tex]
[tex] = \frac{1}{2} - {( \frac{ \sqrt{3} }{2}) }^{2} \times {( \frac{1}{2} )}^{2} [/tex]
[tex] = \frac{1}{2} - \frac{ {( \sqrt{3}) }^{2} }{ {2}^{2} } \times \frac{ {1}^{2} }{ {2}^{2} } [/tex]
[tex] = \frac{1}{2} - \frac{3}{4} \times \frac{1}{4} [/tex]
[tex] = \frac{1}{2} - \frac{3 \times 1}{4 \times 4} [/tex]
[tex] = \frac{1}{2} - \frac{3}{16} [/tex]
[tex] = \frac{8}{16} - \frac{3}{16} [/tex]
[tex] = \frac{8 - 3}{16} [/tex]
[tex] = \frac{5}{16} [/tex]
[tex]d)sin \: 60^{\circ} - 2sin \: 30^{\circ}cos \: 30^{\circ}[/tex]
[tex] = \frac{ \sqrt{3} }{2} - 2 \times \frac{1}{2} \times \frac{ \sqrt{3} }{2} [/tex]
[tex] = \frac{ \sqrt{3} }{2} - \frac{2 \times 1}{2} \times \frac{ \sqrt{3} }{2} [/tex]
[tex] = \frac{ \sqrt{3} }{2} - \frac{2}{2} \times \frac{ \sqrt{3} }{2} [/tex]
[tex] = \frac{ \sqrt{3} }{2} - 1 \times \frac{ \sqrt{3} }{2} [/tex]
[tex] = \frac{ \sqrt{3} }{2} - \frac{ \sqrt{3} }{2} [/tex]
[tex] = 0[/tex]
