Saturday, 29 June 2019

CBSE Class 10 - Maths - Trigonometry Question - If a sec๐œƒ + b tan๐œƒ + c = 0 and p sec๐œƒ + q tan๐œƒ + r = 0, prove that: (๐‘๐‘Ÿ − ๐‘ž๐‘)² − (๐‘๐‘ − ๐‘Ž๐‘Ÿ)² = (๐‘Ž๐‘ž − ๐‘๐‘)²(#eduvictors)

CBSE Class 10 Maths: Trigonometry Question

CBSE Class 10 - Maths - Trigonometry Question - If a sec๐œƒ + b tan๐œƒ + c = 0 and      p sec๐œƒ + q tan๐œƒ + r = 0,  prove that:  (๐‘๐‘Ÿ − ๐‘ž๐‘)² − (๐‘๐‘ − ๐‘Ž๐‘Ÿ)² = (๐‘Ž๐‘ž − ๐‘๐‘)²(#eduvictors)

Question: If a sec๐œƒ + b tan๐œƒ + c = 0 and  p sec๐œƒ + q tan๐œƒ + r = 0,  prove that:  
                                  (๐‘๐‘Ÿ − ๐‘ž๐‘)² − (๐‘๐‘ − ๐‘Ž๐‘Ÿ)² = (๐‘Ž๐‘ž − ๐‘๐‘)²

Answer:

a sec๐œƒ + b tan๐œƒ + c = 0 
c = - (a sec๐œƒ + b tan๐œƒ)

and p sec๐œƒ + q tan๐œƒ + r = 0
⇒  r = -(p sec๐œƒ + q tan๐œƒ)

LHS = (๐‘๐‘Ÿ − ๐‘ž๐‘)² − (๐‘๐‘ − ๐‘Ž๐‘Ÿ)²

= [b(-p sec๐œƒ - q tan๐œƒ) - q(-a sec๐œƒ - b tan๐œƒ)]² - [p(-a sec๐œƒ - b tan๐œƒ) - a(-p sec๐œƒ - q tan๐œƒ)]²

= [-bp sec๐œƒ - bq tan๐œƒ + aq sec๐œƒ + bq tan๐œƒ ]² - [-ap sec๐œƒ - bp tan๐œƒ + ap sec๐œƒ + aq tan๐œƒ]²

= [aq sec๐œƒ - bp sec๐œƒ]² - [aq tan๐œƒ - bp tan๐œƒ]²

= (aq - bp)²sec²๐œƒ - (aq - bp)²tan²๐œƒ

= (aq - bp)²[sec²๐œƒ - tan²๐œƒ]

= (aq - bp)²                 [∵ sec²๐œƒ - tan²๐œƒ = 1]

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