Question

Following question contains statements given in two columns, which have to be matched. The statements in List-I are labelled as A, B, C and D while the statements in List-II are labelled as p,q,r and s. Any given statement in List-I can have correct matching with ONE statement in List-II.

Answer 1

A) $cosec{10}^{\circ }-\sqrt{3}\sec {10}^{\circ }=\frac{\cos {10}^{\circ }-\sqrt{3}\sin {10}^{\circ }}{\sin {10}^{\circ }\cos {10}^{\circ }}$
$=\frac{2(\frac{1}{2}\cos {10}^{\circ }-\frac{\sqrt{3}}{2}\sin {10}^{\circ })}{\frac{2\sin {10}^{\circ }\cos {10}^{\circ }}{2}}=\frac{4\cos {70}^{\circ }}{\sin {20}^{\circ }}=4$
B) $4\cos {20}^{\circ }-\sqrt{3}\cot {20}^{\circ }=\frac{4\cos {20}^{\circ }\sin {20}^{\circ }-\sqrt{3}\cos {20}^{\circ }}{\sin {20}^{\circ }}$
$=\frac{2\sin {40}^{\circ }-\sqrt{3}\cos {20}^{\circ }}{\sin {20}^{\circ }}=\frac{2\sin {90}^{\circ }\sin {40}^{\circ }-2\cos {30}^{\circ }\cos {20}^{\circ }}{\sin {20}^{\circ }}$
$=\frac{\cos {50}^{\circ }-\cos {130}^{\circ }-\cos {50}^{\circ }-\cos {10}^{\circ }}{\sin {20}^{\circ }}=-\frac{2\cos {70}^{\circ }\cos {60}^{\circ }}{\sin {20}^{\circ }}=-1$
C) $\frac{2\cos {40}^{\circ }-\cos {20}^{\circ }}{\sin {20}^{\circ }}=\frac{\cos {40}^{\circ }+\cos {40}^{\circ }-\cos {20}^{\circ }}{\sin {20}^{\circ }}$
$=\frac{\cos {40}^{\circ }-2\sin {30}^{\circ }\sin {10}^{\circ }}{\sin {20}^{\circ }}=\frac{\cos {40}^{\circ }-\sin {10}^{\circ }}{\sin {20}^{\circ }}$
$=\frac{2\cos {30}^{\circ }\sin {20}^{\circ }}{\sin {20}^{\circ }}=\sqrt{3}$
D) $2\sqrt{2}\sin {10}^{\circ }[\frac{\sec 5^{\circ }}{2}+\frac{\cos {40}^{\circ }}{\sin 5^{\circ }}-2\sin {35}^{\circ }]$
$=2\sqrt{2}\sin {10}^{\circ }[\frac{1}{2\cos 5^{\circ }}+\frac{\cos {40}^{\circ }}{\sin 5^{\circ }}-2\sin {35}^{\circ }]$
$=2\sqrt{2}\sin {10}^{\circ }\frac{\sin 5^{\circ }+2\cos 5^{\circ }\cos {40}^{\circ }-4\sin {35}^{\circ }\sin 5^{\circ }\cos 5^{\circ }}{2\cos 5^{\circ }\sin 5^{\circ }}$
$=2\sqrt{2}[\sin 5+\cos {45}^{\circ }+\cos {35}^{\circ }-\cos {25}^{\circ }+\cos {45}^{\circ }]$

$=2\sqrt{2}[\sin 5^{\circ }2\cos {45}^{\circ }-2\sin {30}^{\circ }\sin 5^{\circ }]=4$