Question

Evaluate:$\frac{\cos {32}^{\circ }\sin {58}^{\circ }+\sin {32}^{\circ }\cos {58}^{\circ }}{{cot}^2{20}^{\circ }+{\tan }^2{70}^{\circ }}$

Answer 1

$\frac{\cos {32}^{\circ }\sin {58}^{\circ }+\sin {32}^{\circ }\cos {58}^{\circ }}{{\cot }^2{20}^{\circ }-{\tan }^2{70}^{\circ }}$

$=\frac{\sin ({32}^{\circ }+{58}^{\circ })}{{\cot }^2({90}^{\circ }-{70}^{\circ })+{\tan }^2{70}^{\circ }}$

$=\frac{\sin {90}^{\circ }}{{\tan }^2{70}^{\circ }+{\tan }^2{70}^{\circ }}$

$=\frac{1}{2{\tan }^2{70}^{\circ }}$

$=\frac{1}{2}{\cot }^2{70}^{\circ }$