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Solve the following equation for x: tan−1(x + 1) + tan−1(x − 1) = tan−1831 - Mathematics

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Question

Solve the following equation for x:

tan−1(x + 1) + tan−1(x − 1) = tan−1`8/31`

Sum
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Solution

Given: tan−1(x + 1) + tan−1(x − 1) = tan−1`8/31`

Take LHS

tan−1(x + 1) + tan−1(x − 1) = tan−1`8/31`

We know that, Formula

tan−1 x + tan-1 y = tan-1 `(x + y)/(1 - xy)`

Thus,

`=> tan^-1  ((x + 1)+(x - 1))/(1 -(x + 1)xx(x - 1)) = tan^-1  8/31`

`=> tan^-1  (2x)/(1-(x^2 - 1)) = tan^-1  8/31`

`=> tan^-1  (2x)/(1 - x^2 + 1) = tan^-1  8/31`

`=> (2x)/(1 - x^2 + 1) = 8/31`

⇒ 62x = 8 − 8x2 + 8

⇒ 4x2 + 62x − 16 = 0

⇒ 6x2 + 31x − 8 = 0

⇒ 4x(x + 8) − 1(x + 8) = 0

⇒ (4x − 1)(x + 8) = 0

⇒ 6x + 1 = 0 or x − 1 = 0

⇒ x = `1/4` or x = −8

Since,

x = `1/4` ∈ `(-sqrt2, sqrt2)`

So, 

x = `1/4` is the root of the given equation

Therefore, 

x = `1/4`

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Chapter 4: Inverse Trigonometric Functions - Exercise 4.11 [Page 82]

APPEARS IN

RD Sharma Mathematics [English] Class 12
Chapter 4 Inverse Trigonometric Functions
Exercise 4.11 | Q 3.02 | Page 82

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