Question

The value of ${\log }_{10}\tan {40}^{\circ }.{\log }_{10}\tan {41}^{\circ }...{\log }_{10}\tan {50}^{\circ }$ is

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

Answer 1

The correct option is A $0$

Given ${\log }_{10}\tan {40}^{\circ }.{\log }_{10}\tan {41}^{\circ }...{\log }_{10}\tan {50}^{\circ }$

$={\log }_{10}\tan {40}^{\circ }.{\log }_{10}\tan {41}^{\circ }\tan {42}^{\circ }\tan {43}^{\circ }\tan {44}^{\circ }\tan {45}^{\circ }\tan {46}^{\circ }...{\log }_{10}\tan {50}^{\circ }$

but $\tan {45}^{\circ }=1\Rightarrow$ $\log \tan {45}^{\circ }=\log 1=0$

$\therefore$ $={\log }_{10}\tan {40}^{\circ }.{\log }_{10}\tan {41}^{\circ }\tan {42}^{\circ }\tan {43}^{\circ }\tan {44}^{\circ }\tan {45}^{\circ }\tan {46}^{\circ }...{\log }_{10}\tan {50}^{\circ }=0$

Ans is $Opt:\left[A\right]$