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


Transcript

Misc 2 (Method 1) Evaluate |■8(cos⁡〖α cosβ〗&cos⁡〖α sinβ〗&−sin⁡α@−sinβ&cosβ&0@sin⁡α cosβ&sin⁡〖α sin⁡β 〗&cos⁡α )| |■8(cos⁡〖α cosβ〗&cos⁡〖α sinβ〗&−sin⁡α@−sinβ&cosβ&0@sin⁡α cosβ&sin⁡〖α sin⁡β 〗&cos⁡α )| Expanding Determinant along C1 = cos α cos β |■8(cos⁡𝛽&0@sin⁡〖𝛼 sin⁡𝛽 〗&cos⁡𝛼 )| – cos α sin β |■8(〖−sin〗⁡𝛽&0@sin⁡〖𝛼 𝑐𝑜𝑠⁡𝛽 〗&cos⁡𝛼 )| – sin α |■8(sin⁡𝛽&cos⁡𝛽@sin⁡〖𝛼 𝑐𝑜𝑠⁡𝛽 〗&sin⁡〖𝛼 sin⁡𝛽 〗 )| = cos α cos β (cos β cos α – 0) – cos α sin β (– sin β cos α – 0) – sin α (– sin2 β sin α – cos2 β sin α) = cos α cos β ( cos β cos α) – cos α sin β ( – sin β cos α) + sin2 α sin2 β + cos2 β sin2 α = cos2 α cos2 β + cos2 α sin2 β + sin2 α ( sin2 β + cos2 β) = cos2 α (1) + sin2 α (1) = cos 2 α + sin2α = 1 Misc 2 (Method 2) Evaluate |■8(cos⁡〖α cosβ〗&cos⁡〖α sinβ〗&−sin⁡α@−sinβ&cosβ&0@sin⁡α cosβ&sin⁡〖α sin⁡β 〗&cos⁡α )| Let ∆ = |■8(cos⁡〖α cosβ〗&cos⁡〖α sinβ〗&−sin⁡α@−sinβ&cosβ&0@sin⁡α cosβ&sin⁡〖α sin⁡β 〗&cos⁡α )| = |■8(𝐜𝐨𝐬⁡〖𝛂 〗 cosβ&𝐜𝐨𝐬⁡〖𝛂 〗 sin β&(−𝒄𝒐𝒔⁡𝜶)/cos⁡𝛼 sin⁡α@−sin β&cos β&0@𝐬𝐢𝐧⁡𝛂 cosβ&𝐬𝐢𝐧⁡𝛂 sin⁡β&𝒔𝒊𝒏⁡𝜶/sin⁡𝛼 〖.cos〗⁡α )| Taking common cos α from R1 & sin α from R3 = cos α.sin β |■8(cos⁡𝛽&sin⁡𝛽&−tan⁡𝛼@−sinβ&cosβ&0@cosβ&sin⁡𝛽&co𝑡⁡α )| Applying R1→ R1 – R3 = cos α.sin β |■8(cos⁡〖𝛽 〗−𝐜𝒐𝒔⁡𝜷&sin⁡〖𝛽−𝒔𝒊𝒏⁡𝜷 〗&−tan⁡〖𝛼−𝒄𝒐𝒕⁡𝜶 〗@−sinβ&cosβ&0@cosβ&sin⁡𝛽&co𝑡⁡α )| Expanding determinant along R1 = cos α . sin α(0|■8(cos⁡𝛽&0@sin⁡𝛽&cot⁡𝛼 )|−0|■8(〖−sin〗⁡𝛽&0@cos⁡𝛽&cot⁡𝛼 )|−〖(tan〗⁡〖𝛼−cot⁡〖𝛼)|■8(sin⁡𝛽&cos⁡𝛽@cos⁡𝛽&sin⁡𝛽 )|〗 〗 ) = cos α . sin α(0−0−〖(tan〗⁡〖𝛼−cot⁡〖𝛼)|■8(sin⁡𝛽&cos⁡𝛽@cos⁡𝛽&sin⁡𝛽 )|〗 〗 ) = cos α sin α (– (tan α + cotα ) ( – sin2 β – cos2 β )) = cos α sin α ( tan + cot α ) (sin2 β + cos2 β ) = cos α sin α (tan α + cot α ) (1) = cos α sin α (sin⁡𝛼/cos⁡𝛼 + cos⁡𝛼/sin⁡𝛼 ) = cos α sin α ((𝒔𝒊𝒏𝟐 𝜶 + 𝒄𝒐𝒔𝟐 𝜶)/cos⁡〖𝛼 sin⁡𝛼 〗 ) = cos α sin α (𝟏/cos⁡〖𝛼 sin⁡𝛼 〗 ) = 1

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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.