Check sibling questions

Misc 9 - Prove tan-1 root x  = 1/2 cos-1 (1 - x)/(1 + x)

Misc. 9 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 2
Misc. 9 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 3

This video is only available for Teachoo black users

Solve all your doubts with Teachoo Black (new monthly pack available now!)


Transcript

Misc 9 Prove tan−1 √x = 1/2 cos−1 ((1 − x)/(1 + x)), x ∈ [0, 1] Solving R.H.S. 1/2 cos−1 ((1 − x)/(1 + x)) Putting x = tan2 θ = 1/2 cos−1 ((1 − tan2θ)/(1 + tan2θ)) = 1/2 cos−1 ((1 − (sin2 θ)/(cos2 θ))/(1 + (sin2 θ)/(cos2 θ))) = 1/2 cos−1 (((cos2 θ − sin2 θ)/(cos2 θ))/((cos2 θ + sin2 θ)/(cos2 θ))) = 1/2 cos−1 ((cos2 θ − sin2 θ)/(cos2 θ + sin2 θ) ) = 1/2 cos−1 ((𝐜𝐨𝐬𝟐 𝛉 − 𝐬𝐢𝐧𝟐 𝛉)/1 ) = 1/2 cos−1 (cos 2𝛉) = 1/2 × 2θ = θ We assumed that x = tan2 θ √𝑥 = tan θ tan -1 √𝑥 = θ (Using cos 2θ = cos2 θ – sin2 θ) Hence, 1/2 cos−1 ((1 − x)/(1 + x)) = θ 𝟏/𝟐 cos−1 ((𝟏 − 𝐱)/(𝟏 + 𝐱)) = tan−1 √𝐱 Hence, R.H.S. = L.H.S. Hence Proved

Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.