Question

The value of $10^{{\log }_e\cot 1^{\circ }}\cdot 10^{{\log }_e\cot 2^{\circ }}\cdot 10^{{\log }_e\cot 3^{\circ }}\dots ..10^{{\log }_e\cot 89^{\circ }}$ is equal to

• A. 0
• B. 10
• C. $\frac{1}{10}$
• D. 1

Answer 1

Answer: (D)

Given expression $=10^{\ln \cot 1^{\circ }+\ln \cot 2^{\circ }+\dots ..+\ln \cot 89^{\circ }}$
$=10^{\ln \left(\cot 11^{\circ }\cdot \cot 2^{\circ }\cdot \cot 3^{\circ }\dots \dots .\cot 89^{\circ }\right)}$
$=10^{\ln \left(\cot 1^{\circ }\cdot \cot 2^{\circ }\dots \dots \cdot \cot \left(90^{\circ }-2\right)\cdot \cot \left(90^{\circ }-1\right)\right)}$
$=10^{\ln (1)}=10^{\circ }=1$