Question

$\tan 1^{\circ }\tan 2^{\circ }\tan 3^{\circ }...\tan {89}^{\circ }=$

• A. $1$
• B. $-1$
• C. $0$
• D. None of above

Answer 1

Answer: (A)

$1$

$\left(\tan 1^{\circ }\tan {89}^{\circ }\right)\left(\tan 2^{\circ }\tan {88}^{\circ }\right).....\left(\tan {44}^{\circ }\tan {46}^{\circ }\right)\tan {45}^{\circ }$, group pair of terms
$=\left\{\tan 1^{\circ }\tan \right({90}^{\circ }-1^{\circ }\left)\right\}\left\{\tan 2^{\circ }\tan \right({90}^{\circ }-2^{\circ }\left)\right\}......\left\{\tan {44}^{\circ }\tan \right({90}^{\circ }-{44}^{\circ }\left)\right\}\tan {45}^{\circ }$
$=\left(\tan 1^{\circ }\cot 1^{\circ }\right)\left(\tan 2^{\circ }\cot 2^{\circ }\right).....\left(\tan {44}^{\circ }\cot {44}^{\circ }\right)\tan {45}^{\circ }$
$=(1\times 1.....1)\times 1=1$

Note that $\tan \theta \times \cot \theta =1$