Question

cot10. cot20. cot60. cot70. cot80 = ?
with explanation ​

Answer 1

We are asked to find the value of Cot10.Cot20.Cot60.Cot70.Cot80.

Solution:

We know that:

CotΦ = tan(90-Φ)

So:

Cot10 = tan(90-10) = tan80.

cot20 = tan(90-20) = tan70.

Substituting the values of cot10 and cot20 with that of tan:

= tan80.tan70.cot60.cot70.cot.80

Let us rearrange the terms for our convenience:

= cot60.cot70.tan70.cot80.tan80.

Now, we also know that:

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

Substituting the values of cot70 and cot80 with that of tan:

$= cot60.\frac{1}{tan70} .tan70.\frac{1}{tan80} .tan80\\\\= cot60.$

$So, cot10.cot20.cot60.cot70.cot80 = cot60 = \frac{1}{\sqrt{3} }$

Answer 2

To find

$\sf{cot10.cot20.cot60.cot70.cot80}$

We know that,

cot∅ = tan(90-∅)

Here,

$\sf{cot10 = tan(90 - 10)} \\ \\ \implies \: \boxed{ \sf{cot10 = tan80}}......(1)$

Also,

$\sf{cot20 = tan(90 - 70)} \\ \\ \implies \: \boxed{\sf{cot20 = tan70}}...........(2)$

We know that,

$\sf{cot60 = \frac{1}{ \sqrt{3}}}$

Putting equations (1) and (2),we get:

cot60.tan70.tan80.cot70.cot80

Since,cot∅ = 1/tan∅

→cot60.tan70.tan80.1/tan70.1/tan80

→cot60

→ 1/√3