Question

The value of ${\log }_{10}\tan 1^{\circ }+{\log }_{10}\tan 2^{\circ }+{\log }_{10}\tan 3^{\circ }....+{\log }_{10}\tan {89}^{\circ }$

• A. $1$
• B. $89$
• C. $\frac{1}{2}$
• D. $0$

Answer 1

Answer: (D)

$0$

${\log }_{10}\tan 1^{\circ }+{\log }_{10}\tan 2^{\circ }+{\log }_{10}\tan 3^{\circ }+.....+{\log }_{10}\tan {88}^{\circ }+{\log }_{10}\tan {89}^{\circ }$ is
$=({\log }_{10}\tan 1^{\circ }+{\log }_{10}\tan {89}^{\circ }+({\log }_{10}\tan 2^{\circ }+{\log }_{10}\tan {88}^{\circ })+......$
$={\log }_{10}\left(\tan 1^{\circ }\tan {89}^{\circ }\right)+{\log }_{10}\left(\tan 2^{\circ }\tan {88}^{\circ }\right)+.....$
$={\log }_{10}(\tan 1^{\circ }\cot 1^{\circ }+{\log }_{10}(\tan 2^{\circ }\tan {88}^{\circ }+....$
$={\log }_{10}1+{\log }_{10}1+....=0$.