Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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Solve the Following Equation: Tan 3 X + Tan X = 2 Tan 2 X - Mathematics

Sum

Solve the following equation:

\[\tan 3x + \tan x = 2\tan 2x\]
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Solution

Given:
\[\tan3x + \tan x = 2 \tan2x\]

Now,

\[\tan3x - \tan2x = \tan2x - \tan x\]
\[ \Rightarrow \tan x (1 + \tan3x \tan2x) = \tan x(1 + \tan2x \tan x) \left[ \tan \left( A - B \right) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \right] \]
\[ \Rightarrow \tan x (1 + \tan3x\tan2x - 1 - \tan2x \tan x) = 0\]
\[ \Rightarrow \tan x \tan2x (\tan3x - \tan x) = 0\]

\[\Rightarrow \tan 2x = 0\] or,
\[\tan x = 0\] or,
\[\tan3x - \tan x = 0\]
And,
\[\tan 2x = 0 \Rightarrow 2x = n\pi \Rightarrow x = \frac{n\pi}{2}, n \in Z\]
or,
\[\tan 3x - \tan x = 0 \Rightarrow \tan 3x = \tan x \Rightarrow 3x = n\pi + x \Rightarrow 2x = n\pi \Rightarrow x = \frac{n\pi}{2}, n \in Z\]
And,
\[\tan x = 0 \Rightarrow x = m\pi, m \in Z\]
∴ \[x = \frac{n\pi}{2}, n \in Z\] or
\[x = m\pi, m \in Z\]
  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 11 Trigonometric equations
Exercise 11.1 | Q 5.3 | Page 22
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