Subscribe to our Youtube Channel - https://you.tube/teachoo

Last updated at Oct. 21, 2020 by Teachoo

Transcript

Example 5 Express tan−1 cosx/(1 − sinx ) , – π/2 < x < 3π/2 in the simplest form Lets first calculate cos x & 1 – sin x We know that cos 2x = 𝐜𝐨𝐬𝟐𝐱 – 𝐬𝐢𝐧𝟐𝐱 Replacing x by 𝑥/2 cos (2x/2) = cos2 x/2 – sin2 x/2 cos x = cos2 x/2 – sin2 x/2 We write cosx/(1 − sinx ) in form of tan We know that sin 2x = 2 sin x cos x Replacing x by 𝑥/2 sin (2𝑥/2) = 2 sin 𝑥/2 cos 𝑥/2 sin x = 2 sin 𝑥/2 cos 𝑥/2 Solving tan−1 (cosx/(1 〖− sin〗x )) = tan−1 [(cos2 x/2 − sin2 x/2)/(1 − (2 〖sin 〗〖x/2 cos〖 x/2〗 〗 ) )] = tan−1 [(cos2 x/2 − sin2 x/2)/(1 − 2 〖sin 〗〖x/2 〖 cos 〗〖x/2〗 〗 )] = tan−1 [(cos2 x/2 − sin2 x/2)/(cos2 x/2 + sin2 x/2 − 2 〖sin 〗〖x/2 cos〖 x/2〗 〗 )] As sin2 x + cos2 x = 1 Replacing x by 𝑥/2 sin2 𝑥/2 + cos2 𝑥/2 = 1 1 = sin2 𝑥/2 + cos2 𝑥/2 Solving tan−1 (cosx/(1 〖− sin〗x )) = tan−1 [(cos2 x/2 − sin2 x/2)/(1 − (2 〖sin 〗〖x/2 cos〖 x/2〗 〗 ) )] = tan−1 [(cos2 x/2 − sin2 x/2)/(1 − 2 〖sin 〗〖x/2 〖 cos 〗〖x/2〗 〗 )] = tan−1 [(cos2 x/2 − sin2 x/2)/(cos2 x/2 + sin2 x/2 − 2 〖sin 〗〖x/2 cos〖 x/2〗 〗 )] = tan−1 [(cos x/2 + sin x/2)(cos x/2 − sin x/2)/(cos x/2 − sin x/2)^2 ] Using (a − b) (a + b) = a2 − b2 Hence , a = cos 𝑥/2 and b = sin 𝑥/2 Using a2 + b2 – 2ab = (a – b)2 Hence , a = cos 𝑥/2 and b = sin 𝑥/2 = tan−1 [(cos x/2 + sin x/2)(cos x/2 − sin x/2)/(cos x/2 − sin x/2)^2 ] = tan−1 [((cos x/2 + sin x/2))/((cos x/2 − sin x/2) )] Dividing by cos 𝑥/2 = tan−1 ((cos〖 𝑥/( 2 )〗/〖𝑐𝑜𝑠 〗〖 𝑥/2〗 + sin〖 𝑥/( 2 )〗/〖𝑐𝑜𝑠 〗〖 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 [(𝟏 + 〖tan 〗〖𝑥/2〗)/(1 − 〖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 Using tan (x + y ) = 𝒕𝒂𝒏〖𝒙 +〖 𝒕𝒂𝒏〗〖𝒚 〗 〗/(𝟏 − 𝒕𝒂𝒏〖𝒙 𝒕𝒂𝒏𝒚 〗 ) Replace x by 𝜋/4 and y by 𝑥/2

Examples

Example 1
Important

Example 2

Example 3 Important Deleted for CBSE Board 2021 Exams only

Example 4 Deleted for CBSE Board 2021 Exams only

Example 5 Important Deleted for CBSE Board 2021 Exams only You are here

Example 6 Important Deleted for CBSE Board 2021 Exams only

Example 7 Deleted for CBSE Board 2021 Exams only

Example 8 Deleted for CBSE Board 2021 Exams only

Example 9 Important

Example 10 Important Deleted for CBSE Board 2021 Exams only

Example 11 Important Deleted for CBSE Board 2021 Exams only

Example 12 Important Deleted for CBSE Board 2021 Exams only

Example 13 Important Deleted for CBSE Board 2021 Exams only

Chapter 2 Class 12 Inverse Trigonometric Functions

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.