Mathematics
If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that x2 - y2 = a2 - b2.
Trigonometric Identities
63 Likes
Answer
Given,
x = a sec θ + b tan θ and y = a tan θ + b sec θ.
∴ x2 - y2 = (a sec θ + b tan θ)2 - (tan θ + b sec θ)2
⇒ x2 - y2 = a2 sec2 θ + b2 tan2 θ + 2ab sec θ tan θ - (a2 tan2 θ + b2 sec 2 θ + 2ab sec θ tan θ)
⇒ x2 - y2 = a2 sec2 θ + b2 tan2 θ + 2ab sec θ tan θ - a2 tan2 θ - b2 sec 2 θ - 2ab sec θ tan θ
⇒ x2 - y2 = a2(sec2 θ - tan2 θ) - b2(sec2 θ - tan2 θ)
⇒ x2 - y2 = a2 - b2.
Hence, proved that x2 - y2 = a2 - b2.
Answered By
30 Likes