sec 70° sin 20° + cos 20° cosec 70° = ?

Question:

sec 70° sin 20° + cos 20° cosec 70° = ?
(a) 0
(b) 1
(c) 2
(d) –2

 

Solution:

$\sec 70^{\circ} \sin 20^{\circ}+\cos 20^{\circ} \operatorname{cosec} 70^{\circ}$

$=\sec \left(90^{\circ}-20^{\circ}\right) \sin 20^{\circ}+\cos 20^{\circ} \operatorname{cosec}\left(90^{\circ}-20^{\circ}\right)$

$=\operatorname{cosec} 20^{\circ} \sin 20^{\circ}+\cos 20^{\circ} \sec 20^{\circ} \quad\left(\because \sec \left(90^{\circ}-\theta\right)=\operatorname{cosec} \theta\right.$ and $\left.\operatorname{cosec}\left(90^{\circ}-\theta\right)=\sec \theta\right)$

$=\frac{1}{\sin 20^{\circ}} \sin 20^{\circ}+\cos 20^{\circ} \frac{1}{\cos 20^{\circ}} \quad\left(\because \operatorname{cosec} \theta=\frac{1}{\sin \theta}\right.$ and $\left.\sec \theta=\frac{1}{\cos \theta}\right)$

$=1+1$

$=2$

Hence, the correct option is (c).

 

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