Misc 3  - Evaluate determinant |cos alpha cos beta - Solving by simplifying det.

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  1. Chapter 4 Class 12 Determinants
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Misc 3 (Method 1) Evaluate cos﷮α cosβ﷯﷮ cos﷮α sinβ﷯﷮− sin﷮α﷯﷮−sinβ﷮cosβ﷮0﷮ sin﷮α﷯cosβ﷮ sin﷮α sin﷮β﷯﷯﷮ cos﷮α﷯﷯﷯ 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 cos﷮𝛽﷯﷮0﷮ sin﷮𝛼 sin﷮𝛽﷯﷯﷮ cos﷮𝛼﷯﷯﷯ – cos α sin β −sin﷮𝛽﷯﷮0﷮ sin﷮𝛼 𝑐𝑜𝑠﷮𝛽﷯﷯﷮ cos﷮𝛼﷯﷯﷯ – sin α 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 3 (Method 2) Evaluate cos﷮α cosβ﷯﷮ cos﷮α sinβ﷯﷮− sin﷮α﷯﷮−sinβ﷮cosβ﷮0﷮ sin﷮α﷯cosβ﷮ sin﷮α sin﷮β﷯﷯﷮ cos﷮α﷯﷯﷯ Let ∆ = cos﷮α cosβ﷯﷮ cos﷮α sinβ﷯﷮− sin﷮α﷯﷮−sinβ﷮cosβ﷮0﷮ sin﷮α﷯cosβ﷮ sin﷮α sin﷮β﷯﷯﷮ cos﷮α﷯﷯﷯ = 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 cos﷮𝛽﷯﷮ sin﷮𝛽﷯﷮− tan﷮𝛼﷯﷮−sinβ﷮cosβ﷮0﷮cosβ﷮ sin﷮𝛽﷯﷮ co𝑡﷮α﷯﷯﷯ Applying R1→ R1 – R3 = cos α.sin β 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 cos﷮𝛽﷯﷮0﷮ sin﷮𝛽﷯﷮ cot﷮𝛼﷯﷯﷯−0 −sin﷮𝛽﷯﷮0﷮ cos﷮𝛽﷯﷮ cot﷮𝛼﷯﷯﷯− (tan﷮𝛼− cot﷮𝛼) sin﷮𝛽﷯﷮ cos﷮𝛽﷯﷮ cos﷮𝛽﷯﷮ sin﷮𝛽﷯﷯﷯﷯﷯﷯ = cos α . sin α 0−0− (tan﷮𝛼− cot﷮𝛼) 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﷯﷯﷮ cos﷮𝛼 𝑠𝑖𝑛 𝛼﷯﷯ (1) = 1

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