Question

Evaluate the following :
$(cos{0}^{\circ }+sin{45}^{\circ }+sin{30}^{\circ })(sin{90}^{\circ }-cos{45}^{\circ }+cos{60}^{\circ })$

Answer 1

$(cos{0}^{\circ }+sin{45}^{\circ }+sin{30}^{\circ })(sin{90}^{\circ }-cos{45}^{\circ }+cos{60}^{\circ })$

$=(1+\frac{1}{\sqrt{2}}+\frac{1}{2})(1-\frac{1}{\sqrt{2}}+\frac{1}{2})$

$=((1+\frac{1}{2})+\frac{1}{\sqrt{2}})((1+\frac{1}{2})-\frac{1}{\sqrt{2}})$

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

We know that,

$(a+b)(a-b)=a^2-b^2$

Therefore,

$(\frac{3}{2}+\frac{1}{\sqrt{2}})(\frac{3}{2}-\frac{1}{\sqrt{2}})$

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

$=\frac{9}{4}-\frac{1}{2}$

$=\frac{9}{4}-(\frac{1}{2}\times \frac{2}{2})$

$=\frac{9}{4}-\frac{2}{4}$

$=\frac{7}{4}$