Tamil Nadu Board of Secondary EducationHSC Science Class 12th

# Find the number of solutions of the equation tan-1(x-1)+tan-1x+tan-1(x+1)=tan-1(3x) - Mathematics

Sum

Find the number of solutions of the equation tan^-1 (x - 1) + tan^-1x + tan^-1(x + 1) = tan^-1(3x)

#### Solution

tan–1(x – 1) + tan1 x + tan1(x + 1)

= tan1(x – 1) + tan1(x + 1) + tan1x

= tan^-1 [(x - 1 + x + 1)/(1 - (x - 1)(x + 1))] + tan^-1x

= tan^-1 [(2x)/(1 - (x^2 - 1))] + tan^-1x

= tan^-1 [(2x)/(1 - x^2 + 1)] + tan^-1x

= tan^-1 [(2x)/(2 - x^2)] + tan^-1x

= tan^-1 [((2x)/(2 - x^2) + x)/(1 - (2x)/(2 - x^2) * x)]

= tan^-1 [((2x + 2x - x^3)/(2 - x^2))/((2 - x^2 - 2x^2)/(2 - x^2))]

= tan^-1 [(4x - x^3)/(2 - 3x^2)]

Given L.H.S. = R.H.S

tan^-1 [(4x - x^3)/(2 - 3x^2)] = tan^-1 3x

(4x - x^3)/(2 - 3x) = 3x

4x – x3 = 6x – 9x3

8x3 = 2x

8x3 – 2x = 0

2x(x2 – 1) = 0

x = 0, x2 = 1

x = ±1

Number of solutions are three (0, 1 – 1)

Concept: Properties of Inverse Trigonometric Functions
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Chapter 4: Inverse Trigonometric Functions - Exercise 4.5 [Page 166]

#### APPEARS IN

Tamil Nadu Board Samacheer Kalvi Class 12th Mathematics Volume 1 and 2 Answers Guide
Chapter 4 Inverse Trigonometric Functions
Exercise 4.5 | Q 10 | Page 166
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