Slide30.JPG

Slide31.JPG
Slide32.JPG Slide33.JPG

Subscribe to our Youtube Channel - https://you.tube/teachoo

  1. Chapter 2 Class 12 Inverse Trigonometric Functions
  2. Serial order wise

Transcript

Misc 10 Prove cot−1 ((√(1 + sin⁡〖x 〗 ) + √(1 − sin⁡x ))/(√(1 +〖 sin〗⁡x ) − √(1 − sin⁡x ))) = 𝑥/2 , x ∈ (0, 𝜋/4) First, finding √(1+sin⁡𝑥 ) & √(1−sin⁡𝑥 ) separately We know that sin 2x = 2 sin x cos x Replace x by 𝑥/2 sin (2𝑥/2) = 2 sin 𝑥/2 cos 𝑥/2 Adding 1 both sides 1 + sin x = 1 + 2 sin 𝑥/2 cos 𝑥/2 1 + sin x = sin2 𝑥/2 + cos2 𝑥/2 + 2sin 𝑥/2 cos 𝑥/2 1 + sin x = (sin 𝑥/2 + cos 𝑥/2)2 √(1+sin⁡𝑥 ) = sin 𝑥/2 + cos 𝑥/2 As sin2 x + cos2 x = 1 sin2 𝑥/2 + cos2 𝑥/2 = 1 We know that sin 2x = 2sin x cos x Replace x by 𝑥/2 sin 2𝑥/2 = 2sin 𝑥/2 cos 𝑥/2 sin x = 2sin 𝑥/2 cos 𝑥/2 Multiply by –1 on both sides And then, Adding 1 both sides 1 – sin x = 1 – 2 sin 𝑥/2 cos 𝑥/2 1 - sin x = cos2 𝑥/2 + sin2 𝑥/2 – 2sin 𝑥/2 cos 𝑥/2 1 – sin x = (cos (𝑥 )/2 – sin 𝑥/2)2 √(1 −sin⁡𝑥 ) = (cos (𝑥 )/2 – sin 𝑥/2) As sin2 x + cos2 x = 1 sin2 𝑥/2 + cos2 𝑥/2 = 1 Therefore, cot−1 ((√(1 + sin⁡x ) + √(1 − sin⁡x ))/(√(1 + sin⁡x ) − √(1 −〖 sin〗⁡x ))) = cot−1 ((〖sin 〗⁡〖𝑥/2〗 + 〖cos 〗⁡〖𝑥/2〗 + 〖cos 〗⁡〖𝑥/2〗 − 〖sin 〗⁡〖𝑥/2〗 )/(〖sin 〗⁡〖𝑥/2〗 + 〖cos 〗⁡〖𝑥/2〗 − (〖cos 〗⁡〖𝑥/2〗 − 〖sin 〗⁡〖𝑥/2〗 ) )) = cot−1 ((〖sin 〗⁡〖𝑥/2〗 − 〖sin 〗⁡〖𝑥/2〗 + 〖cos 〗⁡〖𝑥/2〗 + 〖cos 〗⁡〖𝑥/2〗)/(〖cos 〗⁡〖𝑥/2〗 − 〖cos 〗⁡〖𝑥/2〗 + 〖sin 〗⁡〖𝑥/2〗 + 〖sin 〗⁡〖𝑥/2〗 )) = cot−1 ((2 cos x/2 )/〖2 sin〗⁡〖 x/2 〗 ) = cot−1 ("cot " x/2) = x/2 = R.H.S. Hence, L.H.S. = R.H.S. Hence Proved

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.