Prove that tan^{- 1} 1/5 + tan^{- 1} 1/7 + tan^{- 1} 1/3 + tan^{- 1} 1/8 = π/4

Solution:

Inverse trigonometric functions as a topic of learning are closely related to the basic trigonometric ratios

The domain (θ  value)  and the range(answer) of the trigonometric ratio are changed to the range and domain of the inverse trigonometric function.

Inverse trigonometric functions are the inverse ratio of the basic trigonometric ratios. 

Here the basic trigonometric function of Sin θ = y, can be changed to θ = sin^-1 y

LHS

= tan^{- 1} 1/5 + tan^{- 1} 1/7 + tan^{- 1} 1/3 + tan^{- 1} 1/8

= tan^{- 1} [(1/5 + 1/7)/(1 - 1/5.1/7)] + tan^{- 1} [(1/3 + 1/8)/(1 - (1/3).(1/8)]

= tan^{- 1} (12/34) + tan^{- 1} (11/23)

= tan^{- 1} (6/17) + tan^{- 1} (11/23)

= tan^{- 1} [(6/17 + 11/23)/(1 - 6/17.11/23)]

= tan^{- 1} (325/325)

= tan^{- 1} (1)

= tan^{- 1} π / 4

= RHS

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NCERT Solutions for Class 12 Maths - Chapter 2 Exercise ME Question 8

Prove that tan^{- 1} 1/5 + tan^{- 1} 1/7 + tan^{- 1} 1/3 + tan^{- 1} 1/8 = π/4

Summary:

Hence we have proved by using inverse trigonometric functions that tan^{- 1} 1/5 + tan^{- 1} 1/7 + tan^{- 1} 1/3 + tan^{- 1} 1/8 = π/4