Prove that tan 50 - tan 40 = 2 tan 10

Solution:

tan 50 = tan (40+10)

We know tan (a+b) = (tan + tan b)/(1-tan tan b)

tan 50 = (tan 40 + tan 10)/(1-tan 40 tan 10)

Cross multiply

tan 50 (1-tan 40 tan 10) = tan 40 + tan 10

tan 50 – tan 50 tan 40 tan 10 = tan 40 + tan 10

Since tan 50 = tan (90-40) = cot 40

So cot 40 tan 40 = 1

Therefore tan 50 – tan 10 = tan 40 + tan 10

=> tan 50 – tan 40 = 2 tan 10

Hence proved.

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