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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Misc 9 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 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 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

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.