Question

tan5 tan 25 tan30 tan65 tan85 = ?

Answer 1

Tan 5 = tan(90-85) = cot85
Tan 25 = Tan(90-65) = Cot65

Tan5 Tan25 Tan30 Tan 65 Tan85 
⇒ Cot85 Cot 65 Tan30 Tan65 Tan85
⇒ Tan 30 (Since Cot85*Tan85 = 1 and Cot65*Tan65 = 1)
⇒ $\frac{1}{ \sqrt{3}}$

Answer 2

Answer:

The value of $\tan 5 ^ { \circ } \tan 25 ^ { \circ } \tan 30 ^ { \circ } \tan 65 ^ { \circ } \tan 85 ^ { \circ }$ is $\frac{1}{\sqrt3}$

To find:

The value of $\tan 5 ^ { \circ } \tan 25 ^ { \circ } \tan 30 ^ { \circ } \tan 65 ^ { \circ } \tan 85 ^ { \circ }$

Solution:

Given,  

$\tan 5 ^ { \circ } \tan 25 ^ { \circ } \tan 30 ^ { \circ } \tan 65 ^ { \circ } \tan 85 ^ { \circ }$

$\begin{array} { c } { = \tan 5 ^ { \circ } \tan 25 ^ { \circ } \tan 30 ^ { \circ } \tan ( 90 - 65 ) ^ { \circ } \tan ( 90 - 85 ) ^ { \circ } } \\\\ { = \tan 5 ^ { \circ } \tan 25 ^ { \circ } \tan 30 ^ { \circ } \cot 25 ^ { \circ } \cot 5 ^ { \circ } } \\ { [ \because \tan ( 90 - x ) = \cot x ] } \\\\ { = \tan 5 ^ { \circ } \tan 25 ^ { \circ } \tan 30 ^ { \circ } \frac { 1 } { \tan 25 ^ { \circ } } \frac { 1 } { \tan 5 ^ { \circ } } } \\ { = \tan 30 ^ { \circ } = \frac { 1 } { \sqrt { 3 } } } \end{array}$

Hence, the value of $\tan 5 ^ { \circ } \tan 25 ^ { \circ } \tan 30 ^ { \circ } \tan 65 ^ { \circ } \tan 85 ^ { \circ }$ is $\frac{1}{\sqrt3}$