Question

show that tan 5×tan 25×tan 30×tan 65×tan 85=1/√3​

Answer 1

Answer:

Step-by-step explanation:

Given :

To show that tan 5° × tan 25° × tan 30° × tan 65° × tan 85° = $\frac{1}{\sqrt{3}}$

Solution :

We know that,

tan 30° = $\frac{1}{\sqrt{3}}$

&

tan x = $\frac{1}{cot \ x}$

⇒ cot x = $\frac{1}{tan \ x}$

Proof :

LHS = tan 5° × tan 25° × tan 30° × tan 65° × tan 85°

= tan 5° × tan 25° × tan 30° × tan (90 - 25)° × tan (90 - 5)°

=  tan 5° × tan 25° × tan 30° × cot 25° × cot 5°

=  tan 5° × tan 25° × tan 30° ×  $\frac{1}{tan \ 25} \times \frac{1}{tan \ 5}$

Hence proved.

= tan 30° = $\frac{1}{\sqrt{3}}$ = RHS