Question

Evaluate : $\frac{\cos {58}^{\circ }}{\sin {32}^{\circ }}+\frac{\sin {22}^{\circ }}{\cos {68}^{\circ }}-\frac{\cos {38}^{\circ }\csc {52}^{\circ }}{\tan {18}^{\circ }\tan {35}^{\circ }\tan {60}^{\circ }\tan {72}^{\circ }\tan {55}^{\circ }}$

Answer 1

We know that,

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

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

$\tan (90-\theta )=\cot \theta$

$\csc \theta =\frac{1}{\sin \theta }$

$\frac{\cos {58}^{\circ }}{\sin {32}^{\circ }}+\frac{\sin {22}^{\circ }}{\cos {68}^{\circ }}-\frac{\cos {38}^{\circ }\csc {52}^{\circ }}{\tan {18}^{\circ }\tan {35}^{\circ }\tan {60}^{\circ }\tan {72}^{\circ }\tan {55}^{\circ }}$

$=\frac{\cos (90-32{)}^{\circ }}{\sin {32}^{\circ }}+\frac{\sin (90-68{)}^{\circ }}{\cos {68}^{\circ }}-\frac{\cos (90-52{)}^{\circ }}{\sin {52}^{\circ }\tan (90-72{)}^{\circ }\tan (90-55{)}^{\circ }\tan {60}^{\circ }\tan {72}^{\circ }\tan {55}^{\circ }}$

$=\frac{\sin {32}^{\circ }}{\sin {32}^{\circ }}+\frac{\cos {68}^{\circ }}{\cos {68}^{\circ }}-\frac{\sin {52}^{\circ }}{\sin {52}^{\circ }\cot {72}^{\circ }\cot {55}^{\circ }\tan {60}^{\circ }\tan {72}^{\circ }\tan {55}^{\circ }}$

$=1+1-\frac{\sin {52}^{\circ }\tan {72}^{\circ }\tan {55}^{\circ }}{\sin {52}^{\circ }\tan {60}^{\circ }\tan {72}^{\circ }\tan {55}^{\circ }}$

$=1+1-\frac{1}{\tan {60}^{\circ }}$

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

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

Multiply numerator and denominator by $\sqrt{3}$

$=\frac{6-\sqrt{3}}{3}$