Question

Question 11: Prove tan¯¹ [ √1+x - √1-x/√1+x + 1-x] = π/4 - 1/2 cos¯¹ x, -1/√2≤ x ≤ 1 [Hint: Put x = cos 2θ]

Class 12 - Math - Inverse Trigonometric Functions

Answer 1

 tan¯¹ [ √1+x - √1-x/√1+x + 1-x] = π/4 - 1/2 cos¯¹ x,  -1/√2≤ x ≤ 1
 Let x = cos 2Ф ⇒ Ф = cos x¯¹ / 2

⇒ tan¯¹ [ { (√1+cos2Ф) - (√1-cos2Ф) } / { ( √1+cos2Ф) + (1-cos2Ф) } ]
⇒ tan¯¹ [ (√2cos²Ф - √2sin²Ф) / ( √2cos²Ф + √2sin²Ф) ]
⇒ tan¯¹ [ (√2 cosФ - √2 sinФ) / (√2 cosФ + √2 sinФ) ]

⇒ Dividing numerator and denominator by √2 cosФ

⇒ tan¯¹ (1+ tanФ / 1 - tanФ)
⇒ tan¯¹ ( tan (π/4 - Ф) )
⇒ π/4 - Ф
⇒ π/4 - 1/2 cos¯¹ x

Hope it helps...