Check sibling questions

Misc 10 - Prove cot-1 ( root (1 + sin x) + root (1 - sin x))

Misc. 10 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 2
Misc. 10 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 3 Misc. 10 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 4


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

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.