Question

Evaluate the value of the following expression:

$A=\frac{\cos {58}^{\circ }}{\sin {32}^{\circ }}+\frac{\sin {22}^{\circ }}{\cos {68}^{\circ }}-\frac{\cos {38}^{\circ }cosec{52}^{\circ }}{\tan {18}^{\circ }\tan {72}^{\circ }}$

Answer 1

Given,

$\cos {58}^{\circ }=\sin ({90}^{\circ }-{58}^{\circ })=\sin {32}^{\circ }$

$\frac{\cos {58}^{\circ }}{\sin {32}^{\circ }}=\frac{\sin {32}^{\circ }}{\sin {32}^{\circ }}=1$

$\sin {22}^{\circ }\cos ({90}^{\circ }-{22}^{\circ })=\cos {68}^{\circ }$

$\frac{\sin {22}^{\circ }}{\cos {68}^{\circ }}=\frac{\cos {68}^{\circ }}{\cos {68}^{\circ }}=1$

$\cos {38}^{\circ }cosec{52}^{\circ }=\frac{\cos {38}^{\circ }}{\sin {52}^{\circ }}$

$=\frac{\cos {38}^{\circ }}{\cos ({90}^{\circ }-{38}^{\circ })}$

$=\frac{\cos {38}^{\circ }}{\cos {38}^{\circ }}=1$

$\tan {18}^{\circ }\tan {72}^{\circ }=\tan {18}^{\circ }\cot ({90}^{\circ }-{18}^{\circ }$

$=\tan {18}^{\circ }{\cot }^{\circ }$

$=1$

Hence, $A=\frac{\cos {58}^{\circ }}{\sin {32}^{\circ }}+\frac{\sin {22}^{\circ }}{\cos {68}^{\circ }}-\frac{\cos {38}^{\circ }cosec{52}^{\circ }}{\tan {18}^{\circ }\tan {72}^{\circ }}$

$A=1+1\frac{1}{1}$

$=2-1$

$=1$