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Question 27 - CBSE Class 12 Sample Paper for 2022 Boards (MCQ Based - for Term 1) - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at March 22, 2023 by

Simplest form of tan−1 ((√(1 + cos ⁡x )   + √(1 - cos⁡x ))/(√(1 + cos⁡x )   - √(1  - cos⁡x ))), π < x < 3π/2 is :

(a) π/4 − x/2                (b) 3π/2 − x/2   
(c) − x/2                       (d) π − x/2

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Question 27 Simplest form of tan−1 ((√(1 +〖 cos〗⁡𝑥 ) + √(1 − cos⁡𝑥 ))/(√(1 + cos⁡𝑥 ) − √(1 − cos⁡𝑥 ))), 𝜋 < x < 3𝜋/2 is : (a) 𝜋/4 − 𝑥/2 (b) 3𝜋/2 − 𝑥/2 (c) − 𝑥/2 (d) 𝜋 − 𝑥/2 We know that cos 2x = 2 cos2 x − 1 Replace x by 𝑥/2 cos x = 2 cos2 𝑥/2 − 1 Adding 1 both sides 1 + cos x = 2 cos2 𝑥/2 √(𝟏+𝒄𝒐𝒔⁡𝒙 ) = √𝟐 cos 𝒙/𝟐 We know that cos 2x = 1 − 2 sin2 x 1 − cos 2x = 2 sin2 x Replace x by 𝑥/2 1 − cos x = 2 sin2 𝑥/2 √(𝟏−𝒄𝒐𝒔⁡𝒙 ) = √𝟐 sin 𝒙/𝟐 Therefore, tan−1 ((√(1 +〖 cos〗⁡𝑥 ) + √(1 − cos⁡𝑥 ))/(√(1 + cos⁡𝑥 ) − √(1 − cos⁡𝑥 ))) = tan−1 ((√𝟐 〖𝒄𝒐𝒔 〗⁡〖𝒙/𝟐〗 + √𝟐 〖𝒔𝒊𝒏 〗⁡〖𝒙/𝟐〗 )/(√𝟐 〖𝒄𝒐𝒔 〗⁡〖𝒙/𝟐〗 − √𝟐 〖𝒔𝒊𝒏 〗⁡〖𝒙/𝟐〗 )) = tan−1 ((〖cos 〗⁡〖𝑥/2〗 − 〖sin 〗⁡〖𝑥/2〗 )/(〖cos 〗⁡〖𝑥/2〗 + 〖sin 〗⁡〖𝑥/2〗 )) Dividing by 〖𝒄𝒐𝒔 〗⁡〖𝒙/𝟐〗 inside = tan−1 (((〖cos 〗⁡〖𝑥/2〗 − 〖sin 〗⁡〖𝑥/2〗)/(cos⁡𝑥/2))/((〖cos 〗⁡〖𝑥/2〗 + 〖sin 〗⁡〖𝑥/2〗)/(cos⁡𝑥/2))) = tan−1 ((1 − tan⁡〖 x/2〗)/(1 +〖 tan〗⁡〖 x/2〗 )) = tan−1 ((𝒕𝒂𝒏⁡〖 𝝅/𝟒〗 − tan⁡〖 𝑥/2〗)/(1 + 〖𝐭𝐚𝐧 〗⁡〖𝝅/𝟒 .〖 tan 〗⁡〖𝑥/2〗 〗 )) = tan−1 ("tan " (𝜋/4−𝑥/2)) = 𝝅/𝟒 − 𝒙/𝟐 So, the correct answer is (A)

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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.