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Last updated at May 12, 2021 by Teachoo
Transcript
Misc 10 Prove cot−1 ((√(1 + sin〖x 〗 ) + √(1 − sinx ))/(√(1 +〖 sin〗x ) − √(1 − sinx ))) = 𝑥/2 , x ∈ (0, 𝜋/4) First, finding √(1+sin𝑥 ) & √(1−sin𝑥 ) 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+𝒔𝒊𝒏𝒙 ) = 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 −𝒔𝒊𝒏𝒙 ) = (cos (𝒙 )/𝟐 – sin 𝒙/𝟐) As sin2 x + cos2 x = 1 sin2 𝑥/2 + cos2 𝑥/2 = 1 Therefore, cot−1 ((√(1 + sinx ) + √(1 − sinx ))/(√(1 + sinx ) − √(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
Miscellaneous
Misc. 2 Important
Misc. 3 Deleted for CBSE Board 2022 Exams
Misc. 4 Important Deleted for CBSE Board 2022 Exams
Misc. 5 Deleted for CBSE Board 2022 Exams
Misc. 6 Deleted for CBSE Board 2022 Exams
Misc. 7 Important Deleted for CBSE Board 2022 Exams
Misc. 8 Important Deleted for CBSE Board 2022 Exams
Misc. 9 Important Deleted for CBSE Board 2022 Exams
Misc. 10 Important Deleted for CBSE Board 2022 Exams You are here
Misc. 11 Important Deleted for CBSE Board 2022 Exams
Misc 12 Important Deleted for CBSE Board 2022 Exams
Misc. 13 Important Deleted for CBSE Board 2022 Exams
Misc. 14 Deleted for CBSE Board 2022 Exams
Misc 15 (MCQ) Important Deleted for CBSE Board 2022 Exams
Misc 16 (MCQ) Important Deleted for CBSE Board 2022 Exams
Misc 17 (MCQ) Deleted for CBSE Board 2022 Exams
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