Question 24 of 50intermediate🔧 ApplyLong Answer4 marks

Prove that:

tan⁡(π4+x)tan⁡(π4−x)=(1+tan⁡x1−tan⁡x)2\frac{\tan \left( \frac{\pi}{4} + x \right)}{\tan \left( \frac{\pi}{4} - x \right)} = \left( \frac{1 + \tan x}{1 - \tan x} \right)^2
Correct Answer

To Prove:

tan⁡(π4+x)tan⁡(π4−x)=(1+tan⁡x1−tan⁡x)2\frac{\tan \left( \frac{\pi}{4} + x \right)}{\tan \left( \frac{\pi}{4} - x \right)} = \left( \frac{1 + \tan x}{1 - \tan x} \right)^2

Proof:

Step 1: Find tan⁡(π4+x)\tan \left( \frac{\pi}{4} + x \right)

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

tan⁡(π4+x)=tan⁡π4+tan⁡x1−tan⁡π4tan⁡x\tan \left( \frac{\pi}{4} + x \right) = \frac{\tan \frac{\pi}{4} + \tan x}{1 - \tan \frac{\pi}{4} \tan x}

Since tan⁡π4=1\tan \frac{\pi}{4} = 1,

tan⁡(π4+x)=1+tan⁡x1−tan⁡x\tan \left( \frac{\pi}{4} + x \right) = \frac{1 + \tan x}{1 - \tan x}

Step 2: Find tan⁡(π4−x)\tan \left( \frac{\pi}{4} - x \right)

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

tan⁡(π4−x)=tan⁡π4−tan⁡x1+tan⁡π4tan⁡x\tan \left( \frac{\pi}{4} - x \right) = \frac{\tan \frac{\pi}{4} - \tan x}{1 + \tan \frac{\pi}{4} \tan x}

Since tan⁡π4=1\tan \frac{\pi}{4} = 1,

tan⁡(π4−x)=1−tan⁡x1+tan⁡x\tan \left( \frac{\pi}{4} - x \right) = \frac{1 - \tan x}{1 + \tan x}

Step 3: Calculate L.H.S.

L.H.S.=tan⁡(π4+x)tan⁡(π4−x)\text{L.H.S.} = \frac{\tan \left( \frac{\pi}{4} + x \right)}{\tan \left( \frac{\pi}{4} - x \right)} =1+tan⁡x1−tan⁡x1−tan⁡x1+tan⁡x= \frac{\frac{1 + \tan x}{1 - \tan x}}{\frac{1 - \tan x}{1 + \tan x}} =1+tan⁡x1−tan⁡x×1+tan⁡x1−tan⁡x= \frac{1 + \tan x}{1 - \tan x} \times \frac{1 + \tan x}{1 - \tan x} =(1+tan⁡x1−tan⁡x)2= \left( \frac{1 + \tan x}{1 - \tan x} \right)^2 =R.H.S.= \text{R.H.S.}

∴ Hence proved.

Exercise: EXERCISE 3.3 | Q: 7 | (Page 25)
For More Understanding

Explanation

This question from Exercise 3.3 requires applying the compound angle formula for tangent. The key identities used are tan⁡(A+B)\tan(A+B) and tan⁡(A−B)\tan(A-B), both derived from the context. The value tan⁡(π4)=1\tan \left( \frac{\pi}{4} \right) = 1 is essential for simplification. Students should carefully apply the subtraction formula which has a '++' in the denominator, unlike the addition formula.

Solution Steps

  1. Step 1: Apply tan⁡(A+B)\tan(A+B) formula to expand tan⁡(π4+x)=1+tan⁡x1−tan⁡x\tan \left( \frac{\pi}{4} + x \right) = \frac{1 + \tan x}{1 - \tan x}

  2. Step 2: Apply tan⁡(A−B)\tan(A-B) formula to expand tan⁡(π4−x)=1−tan⁡x1+tan⁡x\tan \left( \frac{\pi}{4} - x \right) = \frac{1 - \tan x}{1 + \tan x}

  3. Step 3: Divide the two expressions and simplify to get the required result

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