Question

The value of $\log \tan {17}^{\circ }+\log \tan {37}^{\circ }+\log \tan {53}^{\circ }+\log \tan {73}^{\circ }$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A $0$

$\log \tan {17}^{\circ }+\log \tan {37}^{\circ }+\log \tan {53}^{\circ }+\log \tan {73}^{\circ }$
$=\log \left(\tan {17}^{\circ }\tan {37}^{\circ }\tan {53}^{\circ }\tan {73}^{\circ }\right)=\log \left(\right(\tan {17}^{\circ }\tan {73}^{\circ }\left)\right(\tan {37}^{\circ }\tan {53}^{\circ }\left)\right)=\log \left(\right(\tan {17}^{\circ }\tan ({90}^{\circ }-{17}^{\circ })\left(\tan {37}^{\circ }\tan \right({90}^{\circ }-{37}^{\circ }\left)\right)=\log \left(\right(\tan {17}^{\circ }\cot {17}^{\circ }\left)\right(\tan {37}^{\circ }\cot {37}^{\circ }\left)\right)=\log 1=0$