Question

The value of $ \tan( 1^{\circ}) +\, \tan(89^\circ)$ is

• A. $\frac {1} {\sin( 1^{\circ})}$
• B. $\frac {2} {\sin( 2^{\circ})}$
• C. $\frac {2} {\sin( 1^{\circ})}$
• D. $\frac {1} {\sin( 2^{\circ})}$

Answer 1

Answer: (B)

$\tan \left(1^{\circ}\right)+\tan \left(89^{\circ}\right)=\tan \left(1^{\circ}\right)+\tan \left(90^{\circ}-1^{\circ}\right) $
$=\tan \left(1^{\circ}\right)+\cot \left(1^{\circ}\right) $
$=\frac{\sin \left(1^{\circ}\right)}{\cos \left(1^{\circ}\right)}+\frac{\cos \left(1^{\circ}\right)}{\sin \left(1^{\circ}\right)}$
$=\frac{\sin ^{2}\left(1^{\circ}\right)+\cos ^{2}\left(1^{\circ}\right)}{\cos \left(1^{\circ}\right) \sin \left(1^{\circ}\right)} $
$=\frac{1}{\sin \left(1^{\circ}\right) \cos \left(1^{\circ}\right)} $
$=\frac{2}{2 \sin \left(1^{\circ}\right) \cos \left(1^{\circ}\right)}=\frac{2}{\sin \left(2^{\circ}\right)}$