Question

Find the value of the sum ${\log }_{10}\tan 1^{\circ }+{\log }_{10}\tan 2^{\circ }+....{\log }_{10}\tan {89}^{\circ }$.

• A. $1$
• B. $0$
• C. $2$
• D. $3$

Answer 1

The correct option is B $0$

$({\log }_{10}\tan 1^{\circ }+{\log }_{10}\tan 2^{\circ }+.....+{\log }_{10}\tan {89}^{\circ })$

$={\log }_{10}(\tan 1^{\circ }\tan 2^{\circ }....\tan {44}^{\circ }\tan {45}^{\circ }\tan {46}^{\circ }.....\tan {89}^{\circ })$

$={\log }_{10}[\left(\tan 1^{\circ }\tan {89}^{\circ }\right)\left(\tan 2^{\circ }\tan {88}^{\circ }\right)......\left(\tan {44}^{\circ }\tan {46}^{\circ }\right)\tan {45}^{\circ }]$

$={\log }_{10}[\left(\tan 1^{\circ }\cot 1^{\circ }\right)\left(\tan 2^{\circ }\cot 2^{\circ }\right).......\left(\tan {44}^{\circ }\cot {44}^{\circ }\right)\tan {45}^{\circ }]$

$={\log }_{10}1=0$ $[\because \tan 1^{\circ }\cot 1^{\circ }=\tan 2^{\circ }\cot 2^{\circ }=.......=\tan {44}^{\circ }\cot {44}^{\circ }=1and\tan {45}^{\circ }=1]$

Ans: 0