Question

Evaluate: (${\text{cosec 31}}^{\circ }-\sec {59}^{\circ }$)(${\text{cosec 31}}^{\circ }+\sec {59}^{\circ }$)
[2 Marks]

Answer 1

$\begin{array}{cc}\text{Since,}\text{cosec}({90}^{\circ }-A)& =\sec A\text{and}\sec ({90}^{\circ }-A)=\text{cosec}A\end{array}$

$\begin{array}{cc}\therefore \text{cosec}{31}^{\circ }-\sec {59}^{\circ }& =\text{cosec}({90}^{\circ }-{59}^{\circ })-\sec {59}^{\circ }\\ & =\sec {59}^{\circ }-\sec {59}^{\circ }\\ & =0\end{array}$
[1.5 Marks]

Therefore,
(${\text{cosec 31}}^{\circ }-\sec {59}^{\circ }$)(${\text{cosec 31}}^{\circ }+\sec {59}^{\circ }$)
= $0\times ({\text{cosec 31}}^{\circ }+\sec {59}^{\circ }$)
= $0$
[0.5 Marks]