Question

Show that
tan3x tan2x Tanx=tan3x-tan2x-tanx​

Answer 1

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✤ Required Answer:

✒ To show:

• tan3x tan2x tanx = tan3x - tan2x - tanx

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✤ How to solve?

⚡First let's know some points and

Some important formulae of Trigonometry:

1. sin (A + B) = sinAcosB + cosAsinB
2. sin (A – B) = sinAcosB – cosAsinB
3. cos(A + B) = cosAsinB – sinAcosB
4. cos(A – B) = cosAsinB + sinAcosB
5. tan (A + B) = tan A + tan B/(1 – tanAtanB)
6. tan (A – B) = tan A – tan B/(1 + tanAtanB)

☃️ So, let's solve this question....

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✤ Solution:

$\large{ \bf{ \purple{Proof:}}}$

We can write,

➝ 3x = x + 2x

➝ tan 3x = tan (x + 2x)

By using formula:

tan (A + B) = tan A + tan B/(1 – tanAtanB)

➝ tan 3x = tan (x + 2x)

➝ tan 3x = tan x + tan 2x / 1 - tanx tan2x

Cross multiplying,

➝ tan 3x(1 - tanx tan2x) = tan x + tan 2x

Expanding the parentheses,

➝ tan 3x - tanx. tan2x. tan3x = tan x + tan 2x

➝ tan 3x - tan2x - tanx = tan x. tan 2x. tan 3x

➝ tan 3x. tan2x. tan x = tan 3x - tan 2x - tan x

☸ Hence, solved !!

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Answer 2

$\tt \: 3x = 2x + x \\ \\ \\ \leadsto \tt tan \: 3x = tan \bigg(x + 2x \bigg) \\ \\ \\ \tt \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \leadsto\: \frac{tan x \: \: tan2x \: \:}{1 - tanx - tan2x} \\ \\ \\ \leadsto \tt \underbrace{ tan \: 3x - \: \: tan \: 2x \: \: tan \: x} = tan \: x + tan \: 2x \\ \\ \\ \\ \leadsto \tt tan \:3x \: \: tan \: 2x \: tan \: x = tan \: 3x - tan \: 2x - tan \: x$