Question 38 of 50advanced🔧 ApplyLong Answer4 marks

Prove that:

tan⁡4x=4tan⁡x(1−tan⁡2x)1−6tan⁡2x+tan⁡4x\tan 4x = \frac{4 \tan x (1 - \tan^2 x)}{1 - 6 \tan^2 x + \tan^4 x}
Correct Answer

Proof:

We need to prove that

tan⁡4x=4tan⁡x(1−tan⁡2x)1−6tan⁡2x+tan⁡4x\tan 4x = \frac{4 \tan x (1 - \tan^2 x)}{1 - 6 \tan^2 x + \tan^4 x}

We can write 4x=3x+x4x = 3x + x

Therefore,

tan⁡4x=tan⁡(3x+x)\tan 4x = \tan(3x + x)

Using the addition formula:

tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}

We get:

tan⁡4x=tan⁡3x+tan⁡x1−tan⁡3xtan⁡x\tan 4x = \frac{\tan 3x + \tan x}{1 - \tan 3x \tan x}

From the context, we know that

tan⁡3x=3tan⁡x−tan⁡3x1−3tan⁡2x\tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}

Substituting this value:

tan⁡4x=3tan⁡x−tan⁡3x1−3tan⁡2x+tan⁡x1−3tan⁡x−tan⁡3x1−3tan⁡2x⋅tan⁡x\tan 4x = \frac{\frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x} + \tan x}{1 - \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x} \cdot \tan x}

Simplifying the numerator:

Numerator=3tan⁡x−tan⁡3x+tan⁡x(1−3tan⁡2x)1−3tan⁡2x=4tan⁡x−4tan⁡3x1−3tan⁡2x=4tan⁡x(1−tan⁡2x)1−3tan⁡2x\text{Numerator} = \frac{3\tan x - \tan^3 x + \tan x(1 - 3\tan^2 x)}{1 - 3\tan^2 x} = \frac{4\tan x - 4\tan^3 x}{1 - 3\tan^2 x} = \frac{4\tan x(1 - \tan^2 x)}{1 - 3\tan^2 x}

Simplifying the denominator:

Denominator=1−3tan⁡2x−tan⁡4x1−3tan⁡2x=1−3tan⁡2x−3tan⁡2x+tan⁡4x1−3tan⁡2x=1−6tan⁡2x+tan⁡4x1−3tan⁡2x\text{Denominator} = 1 - \frac{3\tan^2 x - \tan^4 x}{1 - 3\tan^2 x} = \frac{1 - 3\tan^2 x - 3\tan^2 x + \tan^4 x}{1 - 3\tan^2 x} = \frac{1 - 6\tan^2 x + \tan^4 x}{1 - 3\tan^2 x}

Therefore:

tan⁡4x=4tan⁡x(1−tan⁡2x)1−3tan⁡2x×1−3tan⁡2x1−6tan⁡2x+tan⁡4x=4tan⁡x(1−tan⁡2x)1−6tan⁡2x+tan⁡4x\tan 4x = \frac{4\tan x(1 - \tan^2 x)}{1 - 3\tan^2 x} \times \frac{1 - 3\tan^2 x}{1 - 6\tan^2 x + \tan^4 x} = \frac{4\tan x(1 - \tan^2 x)}{1 - 6\tan^2 x + \tan^4 x}

∴ Hence proved.

Exercise: EXERCISE 3.3 | Q: 23 | (Page 26)
For More Understanding

Explanation

This proof uses the tangent addition formula and the triple angle formula for tangent, both of which are provided in the textbook context. The key insight is expressing 4x4x as (3x+x)(3x + x) and then systematically substituting the known formula for tan⁡3x\tan 3x. The algebraic simplification requires careful handling of fractions and combining like terms.

Solution Steps

  1. Step 1: Express 4x as (3x + x) and apply tangent addition formula

  2. Step 2: Substitute the formula for tan 3x from the context

  3. Step 3: Simplify numerator by combining terms over common denominator

  4. Step 4: Simplify denominator by combining terms over common denominator

  5. Step 5: Divide numerator by denominator and cancel common factors

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