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Question

Find the value of tan 3A - tan2A - tanA is equal to ________


A

tan 3A tan2A tanA

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B

-tan 3A tan 2A tanA

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C

tan A tan2A - tan2A + tan3A - tan3AtanA

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D

tan 3A + tan2A

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Solution

The correct option is A

tan 3A tan2A tanA


We observe that the given expression contains the terms of tan3A,tan2A & tanA.

We can substitute the value of tan3A & tan2A.This is one of the ways.

Let's substitute the values and analyze what happens

tan3A - tan2A - tanA

= 3tanAtan3A13tan2A2tanA1tan2AtanA

= 3tanAtan3A13tan2A - (2tanA+tanAtan3A1tan2A)

= 3tanAtan3A13tan2A - 3tanAtan3A1tan2A

= (3tanA - tan^3A) [(3tanAtan3A)(2tan2A)(13tan2A)(1tan2A)]

We didn't see any pattern.If we want to proceed from this method, we have to substitute the value of tan3A & tan2A in each option and then check it.But this is not the effective method to solve these types of problem.

3A can be written as 2A + A

3A = 2A + A

tan3A = tan(2A + A)

we know the compound angle fromula for tan(A+B)

tan3A = tan2A+tanA1tan2A.tanA

Cross multiplying gives

tan3A - tan3A tan2A tanA = tan2A + tanA

tan3A - tan3A - tanA = tan3A tan2A tanA

Option A is correct.

This can save a lot of time.


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