CBSE Class 10 - Maths - Trigonometry Question - If a sec𝜃 + b tan𝜃 + c = 0 and p sec𝜃 + q tan𝜃 + r = 0, prove that: (𝑏𝑟 − 𝑞𝑐)² − (𝑝𝑐 − 𝑎𝑟)² = (𝑎𝑞 − 𝑏𝑝)²(#eduvictors)

CBSE Class 10 Maths: Trigonometry Question

Question: If a sec𝜃 + b tan𝜃 + c = 0 and  p sec𝜃 + q tan𝜃 + r = 0,  prove that:  
                                  (𝑏𝑟 − 𝑞𝑐)² − (𝑝𝑐 − 𝑎𝑟)² = (𝑎𝑞 − 𝑏𝑝)²

Answer:

a sec𝜃 + b tan𝜃 + c = 0 

⇒ c = - (a sec𝜃 + b tan𝜃)

and p sec𝜃 + q tan𝜃 + r = 0

⇒  r = -(p sec𝜃 + q tan𝜃)

LHS = (𝑏𝑟 − 𝑞𝑐)² − (𝑝𝑐 − 𝑎𝑟)²

= [b(-p sec𝜃 - q tan𝜃) - q(-a sec𝜃 - b tan𝜃)]² - [p(-a sec𝜃 - b tan𝜃) - a(-p sec𝜃 - q tan𝜃)]²

= [-bp sec𝜃 - bq tan𝜃 + aq sec𝜃 + bq tan𝜃 ]² - [-ap sec𝜃 - bp tan𝜃 + ap sec𝜃 + aq tan𝜃]²

= [aq sec𝜃 - bp sec𝜃]² - [aq tan𝜃 - bp tan𝜃]²

= (aq - bp)²sec²𝜃 - (aq - bp)²tan²𝜃

= (aq - bp)²[sec²𝜃 - tan²𝜃]

= (aq - bp)²                 [∵ sec²𝜃 - tan²𝜃 = 1]