Question

Mark the correct alternative in each of the following:
The value of $si{n}^2{75}^{\circ }-si{n}^2{15}^{\circ }$ is

• A. $\frac{1}{2}$
  
• B. $\frac{\sqrt{3}}{2}$
  
• C. 1
  
• D. 0

Answer 1

The correct option is B $\frac{\sqrt{3}}{2}$


$si{n}^2{75}^{\circ }-si{n}^2{15}^{\circ }$
$=si{n}^2{75}^{\circ }-co{s}^2{15}^{\circ }$
Now, $si{n}^2{75}^{\circ }=sin({45}^{\circ }+{30}^{\circ })$
$=sin{45}^{\circ }cos{30}^{\circ }+cos{45}^{\circ }sin{30}^{\circ }$
$=\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}\times \frac{1}{2}$
$=\frac{\sqrt{3}+1}{2\sqrt{2}}$
$cos{75}^{\circ }=cos({45}^{\circ }+{30}^{\circ })$
$=cos{45}^{\circ }cos{30}^{\circ }-sin{45}^{\circ }sin{30}^{\circ }$
$=\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}\times \frac{1}{2}$
$=\frac{\sqrt{3}-1}{2\sqrt{2}}$
Hence, $si{n}^2{75}^{\circ }-co{s}^2{75}^{\circ }$
$={\left(\frac{\sqrt{3}+1}{2\sqrt{2}}\right)}^2-{\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)}^2$
$=\frac{3+1+2\sqrt{3}-3-1+2\sqrt{3}}{8}$
$=\frac{4\sqrt{3}}{8}$
$=\frac{\sqrt{3}}{2}$