Question

Evaluate each of the following

$\frac{sin{30}^o}{sin{45}^o}+\frac{tan{45}^o}{sec{60}^o}-\frac{sin{60}^o}{sin{45}^o}-\frac{sin{30}^o}{sin{90}^o}$

Answer 1

$\frac{sin{30}^o}{sin{45}^o}+\frac{tan{45}^o}{sec{60}^o}-\frac{sin{60}^o}{sin{45}^o}-\frac{sin{30}^o}{sin{90}^o}$

$=\frac{sin{30}^o}{sin{45}^o}+tan{45}^{o}cos{60}^o-\frac{sin{60}^o}{sin{45}^o}-\frac{sin{30}^o}{sin{90}^o}$

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

$=\frac{1}{\sqrt{2}}+\frac{1}{2}-\sqrt{\frac{3}{2}}-\frac{1}{2}$

$=\frac{1-\sqrt{3}}{\sqrt{2}}$