Question

Evaluate the following :
${\left(\frac{sin{35}^{\circ }}{cos{55}^{\circ }}\right)}^2+{\left(\frac{cos{55}^{\circ }}{sin{35}^{\circ }}\right)}^2-2cos{60}^{\circ }$

Answer 1

${\left(\frac{sin{35}^{\circ }}{cos{55}^{\circ }}\right)}^2+{\left(\frac{cos{55}^{\circ }}{sin{35}^{\circ }}\right)}^2-2cos{60}^{\circ }$

$={\left(\frac{sin({90}^{\circ }-{55}^{\circ })}{cos{55}^{\circ }}\right)}^2+{\left(\frac{cos({90}^{\circ }-{35}^{\circ })}{sin{35}^{\circ }}\right)}^2-2cos{60}^{\circ }$

$={\left(\frac{cos{55}^{\circ }}{cos{55}^{\circ }}\right)}^2+{\left(\frac{sin{35}^{\circ }}{sin{35}^{\circ }}\right)}^2-2cos{60}^{\circ }$

$cos(90-\theta )$= $sin\theta$, $sin(90-\theta )=cos\theta$

$=1^2+1^2-2\times \frac{1}{2}=1$ $[\because cos{60}^{\circ }=\frac{1}{2}]$