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 Sept. 4, 2021 by Teachoo
Simplest form of tan−1 ((√(1 + cos x ) + √(1 - cosx ))/(√(1 + cosx ) - √(1 - cosx ))), π < x < 3π/2 is :
Question 27 Simplest form of tan−1 ((√(1 +〖 cos〗𝑥 ) + √(1 − cos𝑥 ))/(√(1 + cos𝑥 ) − √(1 − cos𝑥 ))), 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+𝒄𝒐𝒔𝒙 ) = √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−𝒄𝒐𝒔𝒙 ) = √𝟐 sin 𝒙/𝟐
Therefore,
tan−1 ((√(1 +〖 cos〗𝑥 ) + √(1 − cos𝑥 ))/(√(1 + cos𝑥 ) − √(1 − cos𝑥 )))
= 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〗)/(cos𝑥/2))/((〖cos 〗〖𝑥/2〗 + 〖sin 〗〖𝑥/2〗)/(cos𝑥/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)
Made by
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.
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