Question

tan10tan20tan30tan40tan50tan60tan70tan80=1​

Answer 1

Answer:

tan 80=tan(90-10)=cot 10

Step-by-step explanation:

tan 80=cot 10

tan 70=cot 20

tan 60=cot 30

tan 50=cot 40

tan 10*cot 10*tan 20*cot 20*tan 30*cot 30*tan 40* cot 40=1

because

tan 10*cot 10=1

tan 20*cot 20=1

tan 30*cot 30=1

tan 40* cot 40=1

1*1*1*1=1

Answer 2

Given : $\tan 10\tan 20\tan 30\tan 40\tan 50\tan 60\tan 70\tan 80$

To find : RHS = 1

Solution :

As we know that

$\tan \theta.\tan(90-\theta)=1$

So, we get :

$\tan 10\tan 80=1\\\\\tan 20\tan 70=1\\\\\tan 30\tan 60=1\\\\\tan 40\tan 50=1$

So, LHS becomes :

$\tan 10\tan 80\tan 20\tan 70\tan 30\tan 60\tan 40\tan 50\\\\=1\times 1\times 1\times 1\\\\=1$

Hence, proved.