[CBSE 2008]

To prove: 

Proof:

Consider L.H.S.

=


Now look for the pairs whose angles give sum of 90°.

Here the sum of angles of tan 10° and tan 80° gives 90°.

Also the sum of angles of tan 40° and tan 50° gives 90°.

So, now change  tan 80° into  tan(90-10°) and tan 50° into  tan(90-40°)

=


We know tan(90-θ) = cot θ

tan 30° = 1/√3

=

= 1 +


Since tan θ = 1/cot θ

= 1 +

= 1 + 1 = 2 = R.H.S.

Hence, proved.

Note: In such questions take the pairs whose angles give sum of 90° ,change one of them in the form of 90-θ and substitute remaining known values.
Like in this case tan 50° is changed into tan(90-40)° so that it will can be written as cot 50° and
tan 50° × cot 50° gives us value 1.